# Use integration by substitution to evaluate the integral of x*sqrt(4x + 1) dx.

## Question:

Use integration by substitution to evaluate {eq}\; \int x \sqrt{4x + 1} \, \mathrm{d}x {/eq}.

## Integrals Using Substitution:

In the Substitution method of integration, we use this to solve the integrals, definite or indefinite both. Integration by parts and the partial fraction methods are another few methods that also can be used. The definite integration can certain properties that can be used to simplify.

The given integral is:

{eq}\; \int x \sqrt{4x + 1} \, \mathrm{d}x\\ \mathrm{Apply\:u-substitution:}\:u=4x+1\\ =\int \frac{\sqrt{u}\left(u-1\right)}{16}du\\ =\frac{1}{16}\cdot \int \:\sqrt{u}\left(u-1\right)du\\ \mathrm{Expand}\:\sqrt{u}\left(u-1\right)=\quad u^{\frac{3}{2}}-\sqrt{u}\\ \rightarrow \frac{1}{16}\left(\int \:u^{\frac{3}{2}}du-\int \sqrt{u}du\right)\\ =\frac{1}{16}\left(\frac{2}{5}u^{\frac{5}{2}}-\frac{2}{3}u^{\frac{3}{2}}\right)+c\\ \mathrm{Substitute\:back}\:u=4x+1\\ \rightarrow =\frac{1}{16}\left(\frac{2}{5}\left(4x+1\right)^{\frac{5}{2}}-\frac{2}{3}\left(4x+1\right)^{\frac{3}{2}}\right)+C {/eq}