Exercise Questions

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

Key Concepts

This exercise focuses on the following concepts:

• Integration of basic trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant)

• Standard integral formulas and their application with linear arguments ($ax+b$)

• Trigonometric identities for simplification ($1+{\tan }^{2}x={\sec }^{2}x$, ${\cos }^{2}x$ half-angle, etc.)

• Integration by substitution (u-substitution) for composite functions

• Algebraic simplification of integrands before integration (e.g., $\csc x\tan x=\sec x$)

• Linearity properties of integrals (sum rule and constant multiple rule)

• Verification of integration results by differentiation

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Important Formulas

Below are the key formulas used in this exercise:

Standard Trigonometric Integrals: $\int \sin (ax)dx=-\frac{\cos (ax)}{a}+C$ $\int \cos (ax)dx=\frac{\sin (ax)}{a}+C$ $\int {\sec }^2(ax+b)dx=\frac{\tan (ax+b)}{a}+C$ $\int {\csc }^2(ax+b)dx=-\frac{\cot (ax+b)}{a}+C$ $\int \sec (ax)\tan (ax)dx=\frac{\sec (ax)}{a}+C$

Logarithmic Forms: $\int \tan (ax)dx=-\frac{1}{a}\ln |\cos (ax)|+C=\frac{1}{a}\ln |\sec (ax)|+C$

$\int \cot (ax)dx=\frac{1}{a}\ln |\sin (ax)|+C$

$\int \csc (ax)dx=-\frac{1}{a}\ln |\csc (ax)+\cot (ax)|+C$

Trigonometric Identities: $1+{\tan }^{2}x={\sec }^{2}x\Rightarrow {\tan }^{2}x={\sec }^{2}x-1$

$1+{\cot }^{2}x={\csc }^{2}x\Rightarrow {\cot }^{2}x={\csc }^{2}x-1$ ${\cos }^{2}x=\frac{1+\cos (2x)}{2}$

Substitution Rule: $\int f(g(x))g^{\prime }(x)dx=\int f(u)du\text{where }u=g(x)$

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Summary

This exercise covers the fundamental techniques of integrating trigonometric functions and their combinations. The primary strategies involve:

1. Applying standard integral formulas directly — especially for $\sin (ax+b)$, $\cos (ax+b)$, ${\sec }^2(ax+b)$, ${\csc }^2(ax+b)$, and $\sec (ax)\tan (ax)$, each introducing a factor of $\frac{1}{a}$.
2. Using trigonometric identities to rewrite integrands into integrable forms — such as converting ${\tan }^{2}x$ to ${\sec }^{2}x-1$, or ${\cos }^{2}x$ to $\frac{1+\cos 2x}{2}$.
3. Simplifying products of trig functions — e.g., $\csc x\tan x=\sec x$ — before integrating.
4. Employing u-substitution for composite functions, where the inner function's derivative appears as a factor.

Particular attention is required for linear transformations inside trigonometric functions ($ax+b$), which introduce scaling factors of $\frac{1}{a}$ in the result. Always verify solutions by differentiating the result to recover the original integrand.