Question

# Evaluate the Integral $\int {\mathrm{xsec}}^{2}\left(\mathrm{x}\right)\mathrm{d}x$

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Solution

## Finding the integral:Given, $\int {\mathrm{xsec}}^{2}\left(\mathrm{x}\right)\mathrm{d}x$Integration by parts is done when conventional methods of integration do not work on the integrand.The formula for integration by parts is given as,$\int f\left(x\right)·g\left(x\right)\mathrm{d}x=f\left(x\right)\int g\left(x\right)dx-\int \left(f\text{'}\left(x\right)\int g\left(x\right)\mathrm{d}x\right)\mathrm{d}x$[General tip: When picking the functions to be substituted into the formula, $f\left(x\right)$ must be a function that is easily differentiable or goes to zero when differentiated repeatedly. Likewise, $g\left(x\right)$ must be the function that is more easily integrated]Let, $f\left(x\right)=x$ and $g\left(x\right)={\mathrm{sec}}^{2}\left(x\right)$. Thus,$\begin{array}{cc}\int x{\mathrm{sec}}^{2}\left(x\right)\mathrm{d}x=\mathrm{x}\int {\mathrm{sec}}^{2}\left(x\right)\mathrm{d}x-\int \left[\frac{d}{d\mathrm{x}}x\int {\mathrm{sec}}^{2}\left(x\right)\mathrm{d}x\right]\mathrm{d}x& \\ =\mathrm{xtan}\mathit{\left(}x\mathit{\right)}-\int \mathrm{tan}\mathit{\left(}x\mathit{\right)}\mathrm{d}x& \left[\because \int {\mathrm{sec}}^{2}\left(\mathrm{x}\right)\mathrm{d}x=\mathrm{tan}\mathit{\left(}x\mathit{\right)}\right]\\ =\mathrm{xtan}\left(x\right)-\mathrm{ln}\left|\mathrm{sec}\mathit{\left(}x\mathit{\right)}\right|+\mathrm{C}& \left[\because \int \mathrm{tan}\left(x\right)\mathrm{d}x=\mathrm{ln}\left|\mathrm{sec}\left(x\right)\right|\right]\end{array}$Hence $\int x{\mathrm{sec}}^{2}\left(x\right)\mathrm{d}x=x\mathrm{tan}\left(x\right)-\mathrm{ln}\left|\mathrm{sec}\left(\mathrm{x}\right)\right|+\mathrm{C}$.

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