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Integration

Short Notes

Integration is the inverse of differentiation. It's used to find areas, volumes, and accumulated quantities.

  • Indefinite Integral: \(\int f(x) \, dx = F(x) + C\) where F'(x) = f(x)
  • Definite Integral: \(\int_a^b f(x) \, dx = F(b) - F(a)\) (Fundamental Theorem of Calculus)

Key Theories & Formulas

1. Basic Integration Formulas

Function Integral
\(\int x^n \, dx\) \(\frac{x^{n+1}}{n+1} + C\) (n ≠ -1)
\(\int \frac{1}{x} \, dx\) $\ln
\(\int e^x \, dx\) \(e^x + C\)
\(\int a^x \, dx\) \(\frac{a^x}{\ln a} + C\)
\(\int \sin x \, dx\) \(-\cos x + C\)
\(\int \cos x \, dx\) \(\sin x + C\)
\(\int \sec^2 x \, dx\) \(\tan x + C\)
\(\int \csc^2 x \, dx\) \(-\cot x + C\)
\(\int \sec x \tan x \, dx\) \(\sec x + C\)
\(\int \frac{1}{1+x^2} \, dx\) \(\tan^{-1} x + C\)
\(\int \frac{1}{\sqrt{1-x^2}} \, dx\) \(\sin^{-1} x + C\)

2. Integration Techniques

Substitution (u-substitution)

\[\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du\]

Integration by Parts

$\(\int u \, dv = uv - \int v \, du\)$ LIATE Rule for choosing u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential

Partial Fractions

For rational functions P(x)/Q(x):

  • Factor Q(x)
  • Decompose into simpler fractions
  • Integrate each part

3. Special Integrals

Integral Result
\(\int \frac{dx}{x^2 + a^2}\) \(\frac{1}{a} \tan^{-1}(\frac{x}{a}) + C\)
\(\int \frac{dx}{x^2 - a^2}\) $\frac{1}{2a} \ln
\(\int \frac{dx}{\sqrt{a^2 - x^2}}\) \(\sin^{-1}(\frac{x}{a}) + C\)
\(\int \frac{dx}{\sqrt{x^2 + a^2}}\) $\ln
\(\int \sqrt{a^2 - x^2} \, dx\) \(\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}(\frac{x}{a}) + C\)

4. Properties of Definite Integrals

  • \(\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx\)
  • \(\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx\)
  • \(\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx\)
  • If f(-x) = f(x) (even): \(\int_{-a}^a f(x) \, dx = 2\int_0^a f(x) \, dx\)
  • If f(-x) = -f(x) (odd): \(\int_{-a}^a f(x) \, dx = 0\)
  • \(\int_0^{2a} f(x) \, dx = 2\int_0^a f(x) \, dx\) if f(2a-x) = f(x)

5. Leibniz Rule (Differentiation under Integral)

\[\frac{d}{dx} \int_{a(x)}^{b(x)} f(t) \, dt = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)\]

Example Problems

Problem 1: Evaluate \(\int_0^1 x e^{x^2} \, dx\)

Solution:

  1. Let u = x², du = 2x dx
  2. \(\int_0^1 x e^{x^2} \, dx = \frac{1}{2} \int_0^1 e^u \, du\)
  3. \(= \frac{1}{2}[e^u]_0^1 = \frac{1}{2}(e - 1) = \mathbf{\frac{e-1}{2}}\)

Problem 2: Evaluate \(\int \frac{x}{x^2 + 4} \, dx\)

Solution:

  1. Let u = x² + 4, du = 2x dx
  2. \(= \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|u| + C\)
  3. \(= \mathbf{\frac{1}{2} \ln(x^2 + 4) + C}\)

Problem 3: Evaluate \(\int_0^\pi x \sin x \, dx\)

Solution (Integration by Parts):

  1. u = x, dv = sin x dx
  2. du = dx, v = -cos x
  3. \(= [-x \cos x]_0^\pi + \int_0^\pi \cos x \, dx\)
  4. \(= [-\pi(-1) - 0] + [\sin x]_0^\pi\)
  5. \(= \pi + 0 = \mathbf{\pi}\)

Hardest GATE Questions

Topic: Integration by Parts with Recursive Pattern

Question (GATE 2018 Style): If \(I_n = \int_0^{\pi/2} \sin^n x \, dx\), find the reduction formula.

Solution:

  1. \(I_n = \int_0^{\pi/2} \sin^{n-1} x \cdot \sin x \, dx\)
  2. Using IBP: u = sin^(n-1)x, dv = sin x dx
  3. \(I_n = [-\sin^{n-1}x \cos x]_0^{\pi/2} + (n-1)\int_0^{\pi/2} \sin^{n-2}x \cos^2 x \, dx\)
  4. \(= 0 + (n-1)\int_0^{\pi/2} \sin^{n-2}x (1-\sin^2 x) \, dx\)
  5. \(= (n-1)I_{n-2} - (n-1)I_n\)
  6. \(I_n + (n-1)I_n = (n-1)I_{n-2}\)
  7. \(I_n = \frac{n-1}{n} I_{n-2}\)

Wallis's Formula:

  • \(I_n = \frac{(n-1)!!}{n!!} \times \begin{cases} \frac{\pi}{2} & n \text{ even} \\ 1 & n \text{ odd} \end{cases}\)

Topic: Leibniz Rule Application

Question (GATE 2016 Variant): If \(F(x) = \int_0^{x^2} e^{t^2} \, dt\), find F'(x).

Solution: Using Leibniz Rule: $\(F'(x) = e^{(x^2)^2} \cdot 2x - e^0 \cdot 0 = \mathbf{2x \cdot e^{x^4}}\)$

Topic: Even/Odd Function Properties

Question: Evaluate \(\int_{-\pi}^{\pi} (x^3 \cos x + x^2 \sin x) \, dx\)

Solution:

  1. Let f(x) = x³cos x + x²sin x
  2. Check: f(-x) = (-x)³cos(-x) + (-x)²sin(-x) = -x³cos x - x²sin x = -f(x)
  3. f(x) is odd!
  4. \(\int_{-\pi}^{\pi} f(x) \, dx = \mathbf{0}\)

References