Integral calculus

Mathematics (B.Sc) · Calculus and analysis · free study notes for exams, placements and interviews

In short

Integral calculus is the study of integration — the inverse of differentiation and, independently, the limit of a sum. Two distinct objects share the name: the indefinite integral (a family of antiderivatives) and the definite integral (a number obtained as a limit of Riemann sums), joined by the Fundamental Theorem of Calculus. It is examined in every B.Sc. Mathematics "Calculus"/"Analysis" paper under UGC CBCS, in GATE MA, CSIR-UGC NET, IIT JAM, ISI/CMI entrances and in the aptitude-plus-maths rounds of analytics and quant employers. Practically, every "total from a rate" computation — distance from speed, charge from current, expected value from a density, interest from a rate — is an integral.

Explained

The two definitions and why both are needed

Antiderivative: F is an antiderivative of f on an interval if F′(x) = f(x). The indefinite integral ∫f(x)dx = F(x) + C; the constant C is essential because two antiderivatives on an interval differ by a constant only (corollary of the Mean Value Theorem).
Riemann integral: partition P: a = x₀ < x₁ < … < xₙ = b, with Δxᵢ = xᵢ − xᵢ₋₁. Lower and upper sums L(P,f) = Σ mᵢΔxᵢ, U(P,f) = Σ MᵢΔxᵢ where mᵢ, Mᵢ are the infimum and supremum of f on [xᵢ₋₁,xᵢ].
f is Riemann integrable iff sup_P L(P,f) = inf_P U(P,f); that common value is ∫ₐᵇ f. Riemann's criterion: integrable iff for every ε>0 some P has U(P,f) − L(P,f) < ε.
Sufficient conditions: f continuous on [a,b] ⟹ integrable; f monotone on [a,b] ⟹ integrable; f bounded with finitely many (indeed, measure-zero) discontinuities ⟹ integrable.
Not integrable: the Dirichlet function (1 on rationals, 0 on irrationals) has L = 0, U = 1 on every partition; any unbounded function on [a,b] is not Riemann integrable (such cases are handled as improper integrals).
Darboux vs Riemann: Darboux uses sup/inf on subintervals, Riemann uses arbitrary tags with mesh → 0; for bounded f on [a,b] the two definitions are equivalent — a standard "state and justify" question.

Properties, FTC and the mean value theorems

Linearity: ∫(αf + βg) = α∫f + β∫g. Additivity: ∫ₐᵇ = ∫ₐᶜ + ∫ꞓᵇ. Orientation: ∫ₐᵇ = −∫_b^a, ∫ₐᵃ = 0.
Monotonicity: f ≤ g on [a,b] ⟹ ∫ₐᵇf ≤ ∫ₐᵇg. Modulus inequality: |∫ₐᵇ f| ≤ ∫ₐᵇ|f| ≤ M(b−a) where M = sup|f|.
FTC Part 1: f integrable on [a,b], F(x) = ∫ₐˣ f(t)dt ⟹ F is continuous on [a,b]; and if f is continuous at c then F′(c) = f(c). So continuity gives an antiderivative constructively.
FTC Part 2 (evaluation): f integrable with an antiderivative F on [a,b] ⟹ ∫ₐᵇ f = F(b) − F(a). This is what turns a limit of sums into arithmetic.
First Mean Value Theorem for integrals: f continuous on [a,b] ⟹ ∃ c ∈ (a,b) with ∫ₐᵇf = f(c)(b−a); f(c) is the average value.
Generalised (weighted) MVT: f continuous, g ≥ 0 integrable ⟹ ∃c with ∫ₐᵇ fg = f(c)∫ₐᵇ g. Bonnet's/second MVT is the companion for monotone g.
Leibniz rule for variable limits: d/dx ∫_{u(x)}^{v(x)} f(t)dt = f(v)v′ − f(u)u′; with an x inside the integrand, differentiate under the sign (needs continuity of ∂f/∂x).
Integrability does not imply the integrand is a derivative everywhere — an integrable f may have no antiderivative (e.g. a step function), and a derivative may fail to be Riemann integrable. Keep the two halves of the FTC separate.

Techniques of integration (the examinable toolkit)

Standard forms: memorise the dozen listed in the formula section; most problems are reduced to these.
Substitution (change of variable): x = g(t), dx = g′(t)dt; for definite integrals change the limits rather than back-substituting. Requires g to be monotone (one-one) on the range used.
Integration by parts: ∫u dv = uv − ∫v du. Choose u by ILATE (Inverse, Log, Algebraic, Trigonometric, Exponential) — the earlier class becomes u.
Partial fractions for rational functions P/Q with deg P < deg Q (divide first if not): linear factor (x−a) → A/(x−a); repeated (x−a)^k → A₁/(x−a) + … + A_k/(x−a)^k; irreducible quadratic → (Ax+B)/(x²+bx+c); repeated quadratic → extra terms with higher powers.
Trigonometric substitution: √(a²−x²) → x = a sinθ; √(a²+x²) → x = a tanθ; √(x²−a²) → x = a secθ.
Weierstrass substitution t = tan(x/2) for rational functions of sin x and cos x: sin x = 2t/(1+t²), cos x = (1−t²)/(1+t²), dx = 2dt/(1+t²).
Special forms: ∫dx/(a + b cos x), ∫(p sin x + q cos x + r)/(a sin x + b cos x + c)dx (write numerator = λ(denominator) + μ(denominator)′ + ν), ∫e^{ax}sin bx dx (by parts twice, or by the standard result).
Reduction formulae: repeated integration by parts giving Iₙ in terms of Iₙ₋₂ — for sinⁿx, cosⁿx, tanⁿx, secⁿx — or in terms of Iₙ₋₁, as for xⁿe^x and (log x)ⁿ.
Properties used as shortcuts: ∫₀ᵃf(x)dx = ∫₀ᵃf(a−x)dx (king's rule); even/odd symmetry; periodic functions ∫₀^{nT} f = n∫₀^T f.
Not every elementary function has an elementary antiderivative — e^{−x²}, sin x/x, 1/log x, √(1+x⁴) (Liouville's theorem). These are evaluated numerically or by series.

Improper integrals, convergence tests and special functions

Type I (infinite limit): ∫ₐ^∞ f = lim_{b→∞} ∫ₐᵇ f. Type II (unbounded integrand): ∫ₐᵇ with f unbounded at a ⟹ lim_{ε→0⁺} ∫_{a+ε}^b f. Split integrals with both defects; each piece must converge separately.
p-tests: ∫₁^∞ dx/x^p converges iff p > 1; ∫₀¹ dx/x^p converges iff p < 1. ∫₀^∞ dx/x^p never converges.
Comparison test: 0 ≤ f ≤ g and ∫g converges ⟹ ∫f converges; ∫f diverges ⟹ ∫g diverges. Limit comparison: if f/g → L ∈ (0,∞), both behave alike.
Absolute convergence ⟹ convergence. ∫₀^∞ (sin x)/x dx = π/2 converges but not absolutely (conditionally convergent) — a favourite objective question.
Cauchy principal value assigns a symmetric limit to some divergent integrals, e.g. PV ∫₋₁¹ dx/x = 0; it is not convergence.
Gamma function: Γ(n) = ∫₀^∞ x^{n−1}e^{−x}dx, n > 0; Γ(n+1) = nΓ(n), Γ(n+1) = n! for integer n ≥ 0, Γ(1/2) = √π.
Beta function: B(m,n) = ∫₀¹ x^{m−1}(1−x)^{n−1}dx = Γ(m)Γ(n)/Γ(m+n), m,n > 0; symmetric, B(m,n) = B(n,m).
Reflection: Γ(n)Γ(1−n) = π/sin(nπ), 0 < n < 1. Trigonometric form: ∫₀^{π/2} sin^p θ cos^q θ dθ = ½B((p+1)/2, (q+1)/2).

Applications in one variable

Area: between curve and x-axis, ∫|f|; between two curves, ∫(upper − lower)dx; in polar coordinates, ½∫r²dθ.
Arc length: for y = f(x), ∫√(1+(y′)²)dx; parametric, ∫√(x′²+y′²)dt; polar, ∫√(r²+(dr/dθ)²)dθ.
Volumes of revolution: disc/washer method π∫y²dx (or π∫(R²−r²)dx); shell method 2π∫x y dx.
Surface of revolution: 2π∫y√(1+(y′)²)dx about the x-axis.
Centroid and moments: x̄ = ∫x·f dx / ∫f dx; second moment ∫x²dA — the link to area moment of inertia used in strength of materials.
Probability: for a continuous random variable with density f, ∫f = 1, P(a<X<b) = ∫ₐᵇ f, E[X] = ∫x f(x)dx. The normal, exponential and gamma densities are all integral identities.
Work, charge and mass: W = ∫F dx, q = ∫i dt, m = ∫ρ dx — all SI: joule, coulomb, kilogram.

Numerical integration (when no closed form exists)

Trapezoidal rule: ∫ₐᵇf ≈ (h/2)[y₀ + 2(y₁+…+y_{n−1}) + yₙ], h = (b−a)/n; error O(h²), exact for linear f.
Simpson's 1/3 rule: (h/3)[y₀ + 4(odd ordinates) + 2(even interior) + yₙ], needs n even; error O(h⁴), exact for cubics.
Simpson's 3/8 rule: (3h/8)[y₀ + 3(y₁+y₂) + 2y₃ + … + yₙ], needs n a multiple of 3.
Gauss–Legendre quadrature: n nodes integrate polynomials up to degree 2n−1 exactly — most accurate per evaluation.
Caution: higher order is not automatically better for non-smooth or oscillatory integrands; Simpson's rule on a function with a kink inside a panel loses its O(h⁴) advantage.

Key points

Indefinite integral - a family F(x)+C of antiderivatives; a function, not a number.
Definite integral - a real number defined as a limit of Riemann sums; sign-aware, so it is not "area" unless f ≥ 0.
Riemann's criterion - integrability ⟺ U(P,f) − L(P,f) can be made arbitrarily small.
Continuity ⟹ integrability, but not conversely; monotone and bounded-with-few-discontinuities functions are also integrable.
FTC-1 - F(x)=∫ₐˣf is continuous always, and differentiable with F′=f wherever f is continuous.
FTC-2 - ∫ₐᵇ f = F(b) − F(a) for any antiderivative F; this is the evaluation tool.
ILATE - the order for choosing u in integration by parts.
King's rule - ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx; collapses many definite integrals in one line.
Symmetry - ∫₋ₐᵃ f = 2∫₀ᵃ f for even f, 0 for odd f; check oddness before computing.
Reduction formula - expresses Iₙ via Iₙ₋₂ (via Iₙ₋₁ for (log x)ⁿ and xⁿe^x); the standard route for sinⁿ, cosⁿ, secⁿ, (log x)ⁿ.
p-test - ∫₁^∞x^{−p} converges iff p>1; ∫₀¹x^{−p} converges iff p<1.
Beta–Gamma link - B(m,n) = Γ(m)Γ(n)/Γ(m+n), with Γ(1/2)=√π.
Conditional convergence - ∫₀^∞(sin x)/x dx converges to π/2 though ∫|sin x|/x diverges.
Non-elementary antiderivatives - e^{−x²}, sin x/x, 1/log x have no elementary closed form.
Average value - f(c) = (1/(b−a))∫ₐᵇ f, guaranteed attained for continuous f.
Leibniz rule - d/dx∫_{u}^{v} f = f(v)v′ − f(u)u′.
Simpson's 1/3 - error O(h⁴) and needs an even number of subintervals; trapezoidal is O(h²).
Discontinuity inside limits - split at the point; blindly applying F(b)−F(a) across a pole gives wrong (often negative) answers.

Formulas and examples

∫xⁿdx = x^{n+1}/(n+1) + C (n ≠ −1); ∫dx/x = ln|x| + C
∫e^{ax}dx = e^{ax}/a + C; ∫a^x dx = a^x/ln a + C
∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C
∫tan x dx = ln|sec x| + C; ∫sec x dx = ln|sec x + tan x| + C
∫dx/(a²+x²) = (1/a)tan⁻¹(x/a) + C; ∫dx/(a²−x²) = (1/2a)ln|(a+x)/(a−x)| + C
∫dx/√(a²−x²) = sin⁻¹(x/a) + C; ∫dx/√(x²+a²) = ln|x+√(x²+a²)| + C
∫√(a²−x²)dx = (x/2)√(a²−x²) + (a²/2)sin⁻¹(x/a) + C
∫u dv = uv − ∫v du (by parts); ∫e^{ax}sin bx dx = e^{ax}(a sin bx − b cos bx)/(a²+b²) + C
Iₙ = ∫₀^{π/2}sinⁿx dx = ((n−1)/n)Iₙ₋₂, with I₀ = π/2, I₁ = 1 (Wallis)
∫₀^{π/2}sin^p θ cos^q θ dθ = ½ B((p+1)/2, (q+1)/2); B(m,n) = Γ(m)Γ(n)/Γ(m+n)
Γ(n) = ∫₀^∞x^{n−1}e^{−x}dx; Γ(n+1) = nΓ(n); Γ(1/2) = √π
Riemann sum: ∫ₐᵇf = lim_{n→∞} Σ_{i=1}^{n} f(a + iΔx)Δx, Δx = (b−a)/n
Arc length s = ∫ₐᵇ√(1+(dy/dx)²)dx (metre); Volume V = π∫ₐᵇy²dx (cubic metre)
Trapezoidal: (h/2)[y₀ + yₙ + 2Σy_interior]; Simpson 1/3: (h/3)[y₀ + yₙ + 4Σy_odd + 2Σy_even]

Symbols: C arbitrary constant of integration; a,b real constants; h step size (b−a)/n; n a positive integer; Γ, B the gamma and beta functions. In applications the units follow the physical quantities — ∫F dx in newton-metre = joule, ∫i dt in ampere-second = coulomb, length in metre, volume in m³.

Example 1 — Riemann sum from first principles. Evaluate ∫₀¹x²dx as a limit of sums.

Δx = 1/n, xᵢ = i/n; Σ f(xᵢ)Δx = Σ_{i=1}^{n}(i²/n²)(1/n) = (1/n³)·n(n+1)(2n+1)/6.
= (2n³+3n²+n)/(6n³) = 1/3 + 1/(2n) + 1/(6n²).
Limit as n → ∞ = 1/3, matching FTC-2: [x³/3]₀¹ = 1/3.

Example 2 — by parts and partial fractions. (a) ∫x²e^x dx: u = x², dv = e^x dx ⟹ x²e^x − 2∫xe^x dx = x²e^x − 2(xe^x − e^x) = e^x(x² − 2x + 2) + C. Check by differentiating: e^x(x²−2x+2) + e^x(2x−2) = x²e^x. ✓ (b) ∫(3x+5)/((x−1)(x+2))dx. Set A/(x−1) + B/(x+2): 3x+5 = A(x+2) + B(x−1). x = 1 → 8 = 3A, A = 8/3; x = −2 → −1 = −3B, B = 1/3. Integral = (8/3)ln|x−1| + (1/3)ln|x+2| + C.

Example 3 — definite integral by king's rule. Evaluate I = ∫₀^{π/2} dx/(1 + tan x) [equivalently ∫₀^{π/2} cos x/(cos x + sin x)dx].

I = ∫₀^{π/2} cos x/(cos x + sin x) dx. Replacing x → π/2 − x: I = ∫₀^{π/2} sin x/(sin x + cos x)dx.
Adding: 2I = ∫₀^{π/2} (cos x + sin x)/(cos x + sin x)dx = ∫₀^{π/2}dx = π/2.
So I = π/4 ≈ 0.7854.

Example 4 — Beta–Gamma and Wallis. Evaluate ∫₀^{π/2} sin⁴θ cos²θ dθ two ways.

Beta form: ½B(5/2, 3/2) = ½·Γ(5/2)Γ(3/2)/Γ(4). Γ(3/2) = ½√π, Γ(5/2) = (3/2)(1/2)√π = (3/4)√π, Γ(4) = 3! = 6.
= ½ · (3/4)√π · (1/2)√π / 6 = ½ · (3π/8)/6 = ½ · π/16 = π/32 ≈ 0.0982.
Wallis check (p, q even): (3·1)(1)/(6·4·2) × π/2 = 3/48 × π/2 = π/32. ✓

Example 5 — numerical integration compared. Estimate ∫₀¹ dx/(1+x²) with n = 4 (exact value tan⁻¹1 = π/4 = 0.785398).

h = 0.25; ordinates y = 1/(1+x²): y₀ = 1, y₁ = 1/1.0625 = 0.941176, y₂ = 1/1.25 = 0.800000, y₃ = 1/1.5625 = 0.640000, y₄ = 0.500000.
Trapezoidal: (0.25/2)[1 + 0.5 + 2(0.941176 + 0.8 + 0.64)] = 0.125[1.5 + 4.762353] = 0.782794 (error 0.00260).
Simpson 1/3: (0.25/3)[1 + 0.5 + 4(0.941176 + 0.64) + 2(0.8)] = 0.0833333[1.5 + 6.324706 + 1.6] = 0.785392 (error 0.0000060).

Integral	Converges?	Reason
∫₁^∞ dx/x²	Yes (=1)	p = 2 > 1
∫₁^∞ dx/x	No	p = 1, grows as ln b
∫₀¹ dx/√x	Yes (=2)	p = 1/2 < 1
∫₀¹ dx/x	No	p = 1 ≥ 1
∫₀^∞ e^{−x}dx	Yes (=1)	exponential decay
∫₀^∞ (sin x)/x dx	Yes (=π/2)	conditionally, not absolutely

Method	Best for	Watch out for
Substitution	Composite functions, f(g(x))g′(x)	Change the limits; keep g one-one
By parts (ILATE)	Products: x·eˣ, x·ln x, inverse-trig	Wrong choice of u loops forever
Partial fractions	Proper rational functions	Divide first if deg P ≥ deg Q
Trig substitution	√(a²±x²), √(x²−a²)	Restrict θ so the root stays positive
Reduction formula	Integer powers sinⁿ, secⁿ, (ln x)ⁿ	State the base cases I₀, I₁
Beta–Gamma	∫₀^{π/2}sin^p cos^q, ∫₀^∞x^{n−1}e^{−x}	Needs p, q > −1; m, n > 0

Where it is used

Analytics and data science (Mu Sigma, Fractal, ZS Associates, Tredence, bank risk teams): probability densities, expected values, CDFs and survival functions are integrals; integration shows up in the maths screening round.
Actuarial and insurance roles (IRDAI-regulated life insurers, Institute of Actuaries of India CM1/CS1): present values, mortality integrals ∫ₜ e^{−δs}·survival ds, and reserve calculations.
Quant and financial engineering (NSE/BSE market-making desks, treasury): Black–Scholes-type pricing is an integral against a normal density; value-at-risk is a tail integral.
Engineering and R&D (ISRO, DRDO, BHEL, L&T design offices): centroids, area moments of inertia, hydrostatic thrust, work done and signal energy ∫v²dt are all definite integrals; MATLAB/SciPy quadrature implements the rules above.
Teaching and academia: B.Sc./M.Sc. lectureship through CSIR-UGC NET and SET; integral calculus with real analysis is the core of the paper.
Software and simulation: numerical quadrature in CFD, finite-element and electronic-design tools; scipy.integrate.quad, numpy.trapz and Simpson routines are standard library calls.

from scipy.integrate import quad
import numpy as np
val, err = quad(lambda x: 1/(1+x**2), 0, 1)   # 0.7853981633974483, err ~ 1e-14
print(val, np.pi/4)

Interview questions

Q: What is the difference between an indefinite and a definite integral? A: The indefinite integral is a family of antiderivatives (a function plus an arbitrary constant); the definite integral is a single number defined as a limit of Riemann sums, independent of any constant.
Q: State the Fundamental Theorem of Calculus and say what each part gives you. A: Part 1: F(x)=∫ₐˣf(t)dt is differentiable with F′=f wherever f is continuous, so continuous functions have antiderivatives. Part 2: ∫ₐᵇf = F(b)−F(a) for any antiderivative F, which lets you evaluate without summing.
Q: Is every Riemann integrable function continuous? A: No — a step function is integrable but discontinuous. Continuity is sufficient, not necessary; boundedness plus a measure-zero discontinuity set suffices.
Q: Give a bounded function on [0,1] that is not Riemann integrable. A: The Dirichlet function (1 on rationals, 0 on irrationals): every subinterval has supremum 1 and infimum 0, so U(P,f)=1 and L(P,f)=0 for all partitions.
Q: Why does ∫₋₁¹ dx/x² not equal −2? A: The integrand is unbounded at x=0, so FTC-2 does not apply; split at 0 and both improper pieces diverge, so the integral diverges.
Q: When does ∫₁^∞ dx/x^p converge, and how does the answer change on (0,1]? A: It converges iff p>1 on [1,∞); on (0,1] ∫₀¹dx/x^p converges iff p<1. No p makes ∫₀^∞ converge.
Q: How do you decide which function to take as u in integration by parts? A: ILATE — inverse trigonometric, logarithmic, algebraic, trigonometric, exponential; the earlier class becomes u because differentiating it simplifies the problem.
Q: Does every elementary function have an elementary antiderivative? A: No. e^{−x²}, sin x/x and 1/ln x are elementary but their antiderivatives are not, by Liouville's theorem; they are evaluated numerically or by series.
Q: What is the practical difference between the trapezoidal and Simpson's 1/3 rule? A: Trapezoidal fits straight lines with error O(h²); Simpson fits parabolas with error O(h⁴) and requires an even number of subintervals, so it is far more accurate for smooth integrands at the same n.
Q: How are the beta and gamma functions related, and why are they useful? A: B(m,n)=Γ(m)Γ(n)/Γ(m+n); together they evaluate ∫₀¹x^{m−1}(1−x)^{n−1}dx and ∫₀^{π/2}sin^p cos^q in one step, and define the gamma and beta probability distributions.
Q: ∫₀^∞(sin x)/x dx = π/2 converges — is it absolutely convergent? A: No. ∫₀^∞|sin x|/x dx diverges, so the convergence is conditional; the cancellation between positive and negative lobes is what makes it converge.
Q: How would you find the average value of a function, and is it always attained? A: Average = (1/(b−a))∫ₐᵇf. For continuous f the Mean Value Theorem for integrals guarantees some c in (a,b) with f(c) equal to it; for a merely integrable f it need not be attained.

Common mistakes

Omitting the constant C in indefinite integrals, or adding one to a definite integral.
Applying F(b)−F(a) across a discontinuity or a pole inside [a,b] — the classic ∫₋₁¹dx/x² = −2 error.
Substituting without changing the limits, or using a substitution that is not one-one on the interval (so the sign of the root flips).
Writing ∫dx/x = ln x instead of ln|x|, and losing the domain restriction.
Treating a definite integral as "area" when f changes sign — the integral is a signed quantity; area needs ∫|f|.
Using Simpson's 1/3 rule with an odd number of subintervals, or forgetting that trapezoidal/Simpson coefficients alternate 4 and 2 only at interior ordinates.
Forgetting to divide a rational function first when deg P ≥ deg Q before partial fractions, and dropping the (Ax+B) numerator for an irreducible quadratic factor.