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 the foundational techniques of indefinite integration (antidifferentiation).

Antiderivatives and Notation

A function $F(x)$ is called an antiderivative of $f(x)$ on an interval if $F^{\prime }(x)=f(x)$ for all $x$ in that interval.

The indefinite integral represents the entire family of antiderivatives: $\int f(x)dx=F(x)+C$ where $C$ is the constant of integration (always required for indefinite integrals).

Notation: $\int$ is the integral sign, $f(x)$ is the integrand, $dx$ is the variable of integration, and $F(x)+C$ is the general antiderivative.

Power Rule for Integration

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

Special case: When $n=-1$, the power rule fails (division by zero). Instead: $\int x^{-1}dx=\int \frac{1}{x}dx=\ln |x|+C$

Applying to radicals and fractions: Convert to fractional/negative exponents first:

• $\sqrt{x}=x^{1/2}$, so $\int \sqrt{x}dx=\frac{x^{3/2}}{3/2}+C=\frac{2}{3}{x}^{3/2}+C$
• $\frac{1}{x^3}=x^{-3}$, so $\int \frac{1}{x^3}dx=\frac{x^{-2}}{-2}+C=-\frac{1}{2x^2}+C$

Exponential Integration (Base $a$)

$\int a^{x}dx=\frac{a^x}{\ln a}+C(a>0,a\ne 1)$

For a linear exponent $\int a^{kx}dx=\frac{a^{kx}}{k\ln a}+C$.

Special case (base $e$): $\int e^{x}dx=e^x+C$ and $\int e^{kx}dx=\frac{1}{k}{e}^{kx}+C$.

Linearity of Integrals

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Important Formulas

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Problem-Solving Strategies

Strategy 1: Algebraic Simplification First

Before integrating, simplify the integrand algebraically:

• Expand products: $\int (1+x)(1-x^2)dx$ → expand to $\int (1+x-x^2-x^3)dx$
• Distribute fractions: $\int \frac{x^2+8}{x^2}dx=\int \left(1+8x^{-2}\right)dx$
• Convert radicals: $\int \sqrt[3]{x^2}dx=\int x^{2/3}dx$

Strategy 2: U-Substitution for Composite Functions

When the integrand contains a composite function $f(g(x))\cdot g^{\prime }(x)$, substitute $u=g(x)$:

Example: $\int x{e}^{x^2}dx$

• Let $u=x^2$, then $du=2xdx$, so $xdx=\frac{du}{2}$
• $\int x{e}^{x^2}dx=\frac{1}{2}\int e^{u}du=\frac{1}{2}{e}^{x^2}+C$

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Summary

Exercise 3.1 covers the fundamental techniques of indefinite integration:

1. Power Rule — the primary tool for integrating $x^n$ ($n\ne -1$)
2. Exponential Rule — for $e^{kx}$ and $a^x$
3. Linearity — sum, difference, and constant multiple rules
4. Algebraic simplification — always simplify before integrating
5. U-substitution — for composite exponential forms

Always include the constant of integration $C$ for every indefinite integral.