Integrate using trigonometric substitution.

{eq}\displaystyle \int x\left(\sqrt{25 + 9x^{2}}\right) dx {/eq}

Question:

Trigonometric Function Substitution to Solve Integration:

• When we have given the function {eq}a^2 - x^2 {/eq} in an integration, then we will substitute {eq}x = a \cos \theta \text{ or } a \sin \theta {/eq} because we know that {eq}\sin^2 \theta + \cos^2 \theta = 1 {/eq}.

• When we have given the function {eq}x^2 - a^2 {/eq} in an integration, then we will substitute {eq}x = a \sec \theta {/eq} because we know that {eq}\sec^2 \theta - 1 = \tan^2 \theta {/eq}.

• When we have given the function {eq}x^2 + a^2 {/eq} in an integration, then we will substitute {eq}x = a \tan \theta {/eq} because we know that {eq}\sec^2 \theta = \tan^2 \theta + 1 {/eq}.

Answer and Explanation: 1