Calculus/Integration techniques/Trigonometric Substitution/Solutions

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1. \int\frac{10}{25x^{2}+1}dx

Let

5x=\tan(\theta)\qquad5dx=\sec^{2}(\theta)d\theta\qquad dx=\frac{\sec^{2}(\theta)}{5}d\theta

Then

\begin{align}\int\frac{10}{25x^{2}+1}dx&=10\int\frac{\sec^{2}(\theta)}{5(\tan^{2}(\theta)+1)}d\theta\\
&=2\int\frac{\sec^{2}(\theta)}{\sec^{2}(\theta)}d\theta\\
&=2\int d\theta=2\theta+C\\
&\mathbf{=2\arctan(5x)+C}\end{align}