# Evaluate the indefinite integral. \int 4x(\tan(x^{2}))\,dx

## Question:

Evaluate the indefinite integral.

{eq}\displaystyle \int 4x(\tan(x^{2}))\,dx {/eq}

## U-Substitution:

If an integral of the form {eq}\displaystyle \int f(g(x))g'(x)\ \mathrm{d}x {/eq} needs to be solved, then we apply the substitution {eq}u = g(x) {/eq}.

U-substitution substitutes {eq}u {/eq}, or any variable, for the inside function of a composite function in an integral.

Clearly, we need to substitute {eq}u {/eq} for {eq}x^2 {/eq} as it is the inner function.

Doing so allows us to also substitute {eq}\mathrm{d}x {/eq} with:

{eq}\begin{align*} \displaystyle u& =x^2 \\ \mathrm{d}u& =2x \ \mathrm{d}x \\ \mathrm{d}x& =\frac{\mathrm{d}u}{2x} \\ \end{align*} {/eq}

Solving for {eq}\displaystyle \int 4x(\tan(x^{2})) \ \mathrm{d}x {/eq} via u-substitution:

{eq}\begin{align*} \displaystyle \int 4x(\tan(x^{2})) \ \mathrm{d}x & =\displaystyle \int 4x(\tan(u)) \ \left(\frac{\mathrm{d}u}{2x}\right) \ \ \ \left[\mathrm{ U-Substitution }\right]\\ & =\displaystyle 2\int \tan u \ \mathrm{d}u\\ & =\displaystyle 2\ln|\sec u|+C\\ \implies \displaystyle \int 4x(\tan(x^{2})) \ \mathrm{d}x& =\displaystyle 2\ln|\sec x^2|+C \ \ \ \left[\mathrm{ Undo \ the \ substitution }\right]\\ \end{align*} {/eq} 