Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

This exercise focuses on integration by substitution (change of variables) — the reverse of the chain rule.

• Integration by substitution (change of variables)

• Standard integral forms involving inverse trigonometric functions
• Completing the square for quadratic expressions

• Integration of composite functions (chain rule in reverse)
• Logarithmic integration (derivative-over-function pattern)

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Important Formulas

Below are the key formulas used in this exercise:

$\int \frac{dx}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+C$

$\int \frac{dx}{\sqrt{a^2-x^2}}={\sin }^{-1}\left(\frac{x}{a}\right)+C$

$\int \frac{f^{\prime }(x)}{f(x)}dx=\ln |f(x)|+C$

$\int [f(x){]}^n{f}^{\prime }(x)dx=\frac{[f(x){]}^{n+1}}{n+1}+C(n\ne -1)$

Substitution rule: If $u=g(x)$, then $\int f(g(x))g^{\prime }(x)dx=\int f(u)du$

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Method: Integration by Substitution

The substitution method transforms a complex integral into a standard form by introducing a new variable $u$.

Steps:

1. Identify a function $u=g(x)$ whose derivative $g^{\prime }(x)$ (or a constant multiple) appears in the integrand.
2. Compute $du=g^{\prime }(x)dx$ and express $dx$ in terms of $du$.
3. Rewrite the entire integral in terms of $u$.
4. Integrate with respect to $u$.
5. Substitute back $u=g(x)$ to express the answer in terms of $x$.

Worked Example 1 — Power substitution:

Evaluate $\int \frac{x^2}{x^3+1}dx$.

Let $u=x^3+1$, so $du=3x^{2}dx\Rightarrow x^{2}dx=\frac{du}{3}$.

$\int \frac{x^2}{x^3+1}dx=\frac{1}{3}\int \frac{du}{u}=\frac{1}{3}\ln |u|+C=\frac{1}{3}\ln |x^3+1|+C$

Worked Example 2 — Inverse trig form:

Evaluate $\int \frac{dy}{y^2+8y+20}$.

Complete the square: $y^2+8y+20=(y+4{)}^2+4$.

Let $u=y+4$, $a=2$:

$\int \frac{du}{u^2+4}=\frac{1}{2}{\tan }^{-1}\left(\frac{u}{2}\right)+C=\frac{1}{2}{\tan }^{-1}\left(\frac{y+4}{2}\right)+C$

Worked Example 3 — ${\sin }^{-1}$ form:

Evaluate $\int \frac{dx}{\sqrt{20-x^2-4x}}$.

Complete the square: $20-x^2-4x=24-(x+2{)}^2$.

Let $u=x+2$, $a=\sqrt{24}=2\sqrt{6}$:

$\int \frac{du}{\sqrt{24-u^2}}={\sin }^{-1}\left(\frac{u}{2\sqrt{6}}\right)+C={\sin }^{-1}\left(\frac{x+2}{2\sqrt{6}}\right)+C$

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Summary

This exercise demonstrates the method of substitution for evaluating indefinite integrals. The key strategy involves identifying a substitution $u=g(x)$ such that $g^{\prime }(x)$ (or a constant multiple) appears in the integrand, transforming the integral into a standard form. Special attention is given to:

• Trigonometric substitutions for quadratic forms (${\tan }^{-1}$ for $x^2+a^2$, ${\sin }^{-1}$ for $\sqrt{a^2-x^2}$)

• Algebraic substitutions for composite power functions and rational functions

• Completing the square to reduce general quadratics to standard forms

• Recognizing the pattern $\int \frac{f^{\prime }(x)}{f(x)}dx=\ln |f(x)|$ for logarithmic integration