# Integrate using trigonometric substitution. {eq}\displaystyle \int x\left(\sqrt{25 + 9x^{2}}\right) dx {/eq}

## Question:

Integrate using trigonometric substitution.

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

## 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}.

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Given:

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

Here we will substitute {eq}\displaystyle x = \frac{5}{3} \tan \theta {/eq}...