Indefinite Integral

Concept Explanation: The indefinite integral of a function \( f(x) \) is the family of all antiderivatives \( F(x) \) such that \( \frac{d}{dx}F(x) = f(x) \). It is denoted by \( \int f(x) \, dx = F(x) + C \), where \( C \) is the constant of integration. Integration is the inverse process of differentiation.

Methods of Integration

• Integration by Substitution: Change the variable of integration from \( x \) to \( t \) where \( x = g(t) \). Then, \[ \int f(x) \, dx = \int f(g(t)) g'(t) \, dt. \]
• Integration by Partial Fractions: For rational functions \( \frac{f(x)}{g(x)} \), if degree of numerator \( \geq \) degree of denominator, perform polynomial division first. Then decompose into partial fractions based on the factorization of denominator and integrate each simpler fraction.
• Integration by Parts: For functions \( U(x) \) and \( V(x) \), \[ \int U \, dV = UV - \int V \, dU. \]

Formula Derivation

For Integration by Parts, starting from the product rule: \[ \frac{d}{dx}(UV) = U \frac{dV}{dx} + V \frac{dU}{dx} \implies U \frac{dV}{dx} = \frac{d}{dx}(UV) - V \frac{dU}{dx}. \] Integrating both sides, \[ \int U \, dV = UV - \int V \, dU. \]

Worked Illustrations

Example 1: Integrate \( \int \tan^4(\sqrt{x}) \sec^2(\sqrt{x}) / \sqrt{x} \, dx \).

Solution: Let \( t = \sqrt{x} \), then \( dt = \frac{1}{2\sqrt{x}} dx \Rightarrow dx = 2t \, dt \).

Rewrite integral: \[ I = \int \tan^4(t) \sec^2(t) \frac{1}{t} \times 2t \, dt = 2 \int \tan^4(t) \sec^2(t) \, dt. \]

Substitute \( u = \tan t \Rightarrow du = \sec^2 t \, dt \), so \[ I = 2 \int u^4 \, du = 2 \frac{u^5}{5} + C = \frac{2}{5} \tan^5(\sqrt{x}) + C. \]

Example 2: Find \( \int \frac{dx}{3x^2 + 13x - 10} \).

Solution: Factor denominator or complete the square: \[ 3x^2 + 13x - 10 = 3 \left( x^2 + \frac{13}{3} x - \frac{10}{3} \right). \]

Complete the square: \[ x^2 + \frac{13}{3} x = \left( x + \frac{13}{6} \right)^2 - \left( \frac{13}{6} \right)^2. \]

Rewrite integral: \[ I = \frac{1}{3} \int \frac{dx}{\left( x + \frac{13}{6} \right)^2 - \left( \frac{17}{6} \right)^2}. \]

Put \( t = x + \frac{13}{6} \), \( dt = dx \), then \[ I = \frac{1}{3} \int \frac{dt}{t^2 - \left( \frac{17}{6} \right)^2} = \frac{1}{3} \int \frac{dt}{(t - \frac{17}{6})(t + \frac{17}{6})}. \]

Use partial fractions: \[ \frac{1}{(t - a)(t + a)} = \frac{A}{t - a} + \frac{B}{t + a}. \]

Solving for \( A, B \), integrate and simplify to get \[ I = \frac{1}{17} \log \left| \frac{3x - 2}{x + 5} \right| + C. \]

Practice Set

• Level 1 – Easy:
  • Find \( \int x^3 \, dx \).
  • Evaluate \( \int \sin x \, dx \).
  • Compute \( \int e^{2x} \, dx \).
• Level 2 – Moderate:
  • Evaluate \( \int x e^{x} \, dx \) using integration by parts.
  • Find \( \int \frac{dx}{x^2 - 4} \) using partial fractions.
  • Calculate \( \int \tan^3 x \, dx \) using substitution.
• Level 3 – Challenging:
  • Evaluate \( \int \frac{x^2 + 3x + 2}{(x+1)(x+2)^2} \, dx \) by partial fractions.
  • Find \( \int x^2 \sin x \, dx \) using integration by parts twice.
  • Compute \( \int \frac{dx}{x^3 + x} \) by partial fraction decomposition.

Answer Key

• \( \int x^3 \, dx = \frac{x^4}{4} + C \)
• \( \int \sin x \, dx = -\cos x + C \)
• \( \int e^{2x} \, dx = \frac{e^{2x}}{2} + C \)
• \( \int x e^{x} \, dx = x e^{x} - e^{x} + C \)
• \( \int \frac{dx}{x^2 - 4} = \frac{1}{4} \log \left| \frac{x-2}{x+2} \right| + C \)
• \( \int \tan^3 x \, dx = \frac{1}{2} \tan^2 x - \log |\cos x| + C \)
• \( \int \frac{x^2 + 3x + 2}{(x+1)(x+2)^2} \, dx = -\frac{1}{x+2} + \log |x+1| + C \)
• \( \int x^2 \sin x \, dx = -x^2 \cos x + 2x \sin x + 2 \cos x + C \)
• \( \int \frac{dx}{x^3 + x} = \frac{1}{2} \log |x| - \frac{1}{2} \log |x^2 + 1| + C \)

Quick Reference

Integral	Result
\( \int x^n \, dx \)	\( \frac{x^{n+1}}{n+1} + C, n \neq -1 \)
\( \int e^{ax} \, dx \)	\( \frac{e^{ax}}{a} + C \)
\( \int \sin(ax) \, dx \)	\( -\frac{1}{a} \cos(ax) + C \)
\( \int \cos(ax) \, dx \)	\( \frac{1}{a} \sin(ax) + C \)
Integration by Parts	\( \int U \, dV = UV - \int V \, dU \)
Integration by Substitution	\( \int f(g(t)) g'(t) \, dt \)

Glossary

• Integral: The antiderivative or primitive of a function.
• Indefinite Integral: Integral without limits, representing a family of functions.
• Integration by Substitution: Method changing variable to simplify integral.
• Partial Fractions: Decomposition of rational functions into simpler fractions.
• Integration by Parts: Technique based on product rule for differentiation.
• Constant of Integration (C): Arbitrary constant added to indefinite integrals.

Definite Integral

Concept Explanation: The definite integral of a function \( f(x) \) from \( a \) to \( b \) is the numeric value \[ \int_a^b f(x) \, dx = F(b) - F(a), \] where \( F(x) \) is any antiderivative of \( f(x) \). It represents the net area under the curve \( y = f(x) \) between \( x = a \) and \( x = b \).

Properties of Definite Integrals

• \( \int_a^a f(x) \, dx = 0 \)
• \( \int_a^b f(x) \, dx = - \int_b^a f(x) \, dx \)
• \( \int_a^b [f(x) + g(x)] \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx \)
• \( \int_a^b k f(x) \, dx = k \int_a^b f(x) \, dx \), where \( k \) is constant
• Splitting integral: \( \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx \)
• Symmetry: For even function \( f(-x) = f(x) \), \( \int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx \)
• For odd function \( f(-x) = -f(x) \), \( \int_{-a}^a f(x) \, dx = 0 \)

Worked Illustrations

Example 3: Evaluate \( I = \int_0^{\pi/4} \sin^3(2t) \cos(2t) \, dt \).

Solution: Let \( u = \sin(2t) \), then \( du = 2 \cos(2t) dt \Rightarrow \cos(2t) dt = \frac{1}{2} du \).

Change limits: when \( t=0 \), \( u=0 \); when \( t=\pi/4 \), \( u=1 \).

Rewrite integral: \[ I = \int_0^{\pi/4} \sin^3(2t) \cos(2t) \, dt = \frac{1}{2} \int_0^1 u^3 \, du = \frac{1}{2} \times \frac{u^4}{4} \Big|_0^1 = \frac{1}{8}. \]

Example 4: Evaluate \( \int_1^2 \frac{x}{(x+1)(x+2)} \, dx \).

Solution: Decompose into partial fractions: \[ \frac{x}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}. \]

Multiply both sides by \( (x+1)(x+2) \): \[ x = A(x+2) + B(x+1) = (A+B)x + (2A + B). \]

Equate coefficients: \[ A + B = 1, \quad 2A + B = 0. \]

Solving gives \( A = -1, B = 2 \).

Rewrite integral: \[ I = \int_1^2 \left( \frac{-1}{x+1} + \frac{2}{x+2} \right) dx = - \int_1^2 \frac{dx}{x+1} + 2 \int_1^2 \frac{dx}{x+2}. \]

Integrate: \[ I = -[\log|x+1|]_1^2 + 2[\log|x+2|]_1^2 = -\log 3 + \log 2 + 2 \log 4 - 2 \log 3 = \log \frac{32}{27}. \]

Practice Set

• Level 1 – Easy:
  • Evaluate \( \int_0^1 x^2 \, dx \).
  • Find \( \int_1^2 \frac{1}{x} \, dx \).
  • Calculate \( \int_0^{\pi} \sin x \, dx \).
• Level 2 – Moderate:
  • Evaluate \( \int_0^1 x e^{x^2} \, dx \) using substitution.
  • Find \( \int_1^3 \frac{2x+1}{x^2 + x} \, dx \) using partial fractions.
  • Calculate \( \int_{-1}^1 x^3 \, dx \) and explain the result.
• Level 3 – Challenging:
  • Evaluate \( \int_0^{\pi/2} \sin^2 x \, dx \) using symmetry and reduction formulas.
  • Find \( \int_0^2 \frac{x^2}{x^3 + 1} \, dx \) using substitution and partial fractions.
  • Prove \( \int_{-a}^a f(x) \, dx = 0 \) if \( f \) is odd.

Answer Key

• \( \int_0^1 x^2 \, dx = \frac{1}{3} \)
• \( \int_1^2 \frac{1}{x} \, dx = \log 2 \)
• \( \int_0^{\pi} \sin x \, dx = 2 \)
• \( \int_0^1 x e^{x^2} \, dx = \frac{e - 1}{2} \)
• \( \int_1^3 \frac{2x+1}{x^2 + x} \, dx = 2 \log 3 - \log 2 \)
• \( \int_{-1}^1 x^3 \, dx = 0 \) because \( x^3 \) is odd.
• \( \int_0^{\pi/2} \sin^2 x \, dx = \frac{\pi}{4} \)
• \( \int_0^2 \frac{x^2}{x^3 + 1} \, dx = \frac{1}{3} \log 9 \)
• Proof: For odd \( f \), \( f(-x) = -f(x) \), so \[ \int_{-a}^a f(x) \, dx = 0. \]

Quick Reference

Property	Formula
Fundamental Theorem	\( \int_a^b f(x) \, dx = F(b) - F(a) \)
Reversing limits	\( \int_a^b f(x) \, dx = - \int_b^a f(x) \, dx \)
Linearity	\( \int_a^b [k f(x) + g(x)] \, dx = k \int_a^b f(x) \, dx + \int_a^b g(x) \, dx \)
Splitting integral	\( \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx \)
Even function	\( \int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx \)
Odd function	\( \int_{-a}^a f(x) \, dx = 0 \)

Glossary

• Definite Integral: Integral with limits, giving numeric value.
• Limits of Integration: The values \( a \) and \( b \) between which the integral is evaluated.
• Antiderivative: A function whose derivative is the integrand.
• Even Function: \( f(-x) = f(x) \), symmetric about y-axis.
• Odd Function: \( f(-x) = -f(x) \), symmetric about origin.
• Linearity: Property allowing splitting and scaling of integrals.