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Calculus

Section 1.1 Inverse Trigonometric Substitution

Inverse trigonometric substitution is a technique used to evaluate integrals that contain expressions of the form \(\sqrt{a^2 - x^2}\text{,}\) \(\sqrt{x^2 - a^2}\text{,}\) or \(\sqrt{a^2 + x^2}\text{.}\) This method uses trigonometric identities to simplify these integrals.

Subsection 1.1.1 Summary of Inverse Trigonometric Substitution

Table 1.1.1.
\(\text{form}\) \(\text{substitution}\) \(\text{conversion}\) \(\text{triangle}\)
\(\sqrt{a^2 - x^2}\) \(x = a \sin{\theta}\) \(\sqrt{a^2 - x^2} \rightarrow a \cos{\theta}\) \(\sqrt{a^2 - x^2}, x, a\)
\(\sqrt{a^2 + x^2}\) \(x = a \tan{\theta}\) \(\sqrt{a^2 + x^2} \rightarrow a \sec{\theta}\) \(x, a, \sqrt{a^2 + x^2}\)
\(\sqrt{x^2 - a^2}\) \(x = a \sec{\theta}\) \(\sqrt{x^2 - a^2} \rightarrow a \tan{\theta}\) \(\sqrt{x^2 - a^2}, a, x\)
  1. Identify the form of the substitution (sine, tangent, or secant).
  2. Apply the substitution. Sketch a right triangle and label the sides, to represent the relationships between \(x, a\) and \(\theta\text{.}\)
  3. Simplify the integral.
  4. Integrate with the new variable \(\theta\text{,}\) using previous strategies, typically a trigonometric integral.
  5. Back-substitute using the inverse trigonometric function to substitute back for \(x\text{.}\)

Subsection 1.1.2 Examples

Checkpoint 1.1.2. Example.

Evaluate \(\int \frac{x^2}{\sqrt{9-x^2}} \,dx\text{.}\)
Hint.
Let \(x = 3 \sin{\theta}\text{,}\) so \(dx = 3 \cos{\theta} \,d\theta\text{.}\)
Answer.
\begin{gather*} \frac{9}{2} \sin^{-1}\brac{\frac{x}{3}} - \frac{1}{2} x \sqrt{9-x^2} + C \end{gather*}
Solution.
\begin{align*} \amp = \int \frac{(3 \sin{\theta})^2}{3 \cos{\theta}} \cdot 3 \cos{\theta} \,d\theta\\ \amp = 9 \int \sin^2{\theta} \,d\theta\\ \amp = ...\\ \amp = \frac{9}{2} \sin^{-1}\brac{\frac{x}{3}} - \frac{1}{2} x \sqrt{9-x^2} + C \end{align*}

Checkpoint 1.1.3. Example.

Determine
\begin{gather*} \int \frac{1}{x^2 \sqrt{4 + x^2}} \,dx \end{gather*}

Checkpoint 1.1.4. Example.

Determine
\begin{gather*} \int \frac{1}{(1 + 9x^2)} \,dx \end{gather*}

Checkpoint 1.1.5. Example.

Determine
\begin{gather*} \int \frac{\sqrt{x^2 - 25}}{x} \,dx \end{gather*}

Checkpoint 1.1.6.

Evaluate
\begin{gather*} \int \frac{\sqrt{x^2 - 1}}{x^4} \,dx \end{gather*}

Checkpoint 1.1.7.

Evaluate
\begin{gather*} \int \frac{\sqrt{25x^2 - 4}}{x} \,dx \end{gather*}

Checkpoint 1.1.8.

Evaluate
\begin{gather*} \int \frac{\sqrt{x^2+16}}{x^2} \,dx \end{gather*}

Checkpoint 1.1.9.

Evaluate
\begin{gather*} \int \frac{x}{\sqrt{9-4x^2}} \,dx \end{gather*}

Checkpoint 1.1.10.

Evaluate
\begin{gather*} \int \frac{1}{\sqrt{25x^2-4}} \,dx \end{gather*}

Checkpoint 1.1.11.

Evaluate
\begin{gather*} \int \frac{1}{x^2 \sqrt{x^2-1}} \,dx \end{gather*}

Checkpoint 1.1.12.

Evaluate
\begin{gather*} \int \frac{8}{(4x^2+1)^2} \,dx \end{gather*}
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