Definite Integration by Substitution

$$ \begin{align} \displaystyle u &= f(x) \\ du &= \dfrac{du}{dx} \times dx \\ \end{align}$$ Converting $x$-values to corresponding $u$-values are required.

Example 1

Find $\displaystyle 2\int_{0}^{1}{\sqrt{2x+1}}dx$. It is not required converting $x$-values to corresponding $u$-values, but substitute $x$-values after integrating it.

Example 2

Find $\displaystyle \int_{0}^{1}{x\sqrt{1+x^2}}dx$.

Example 3

Find $\displaystyle \int_{0}^{1}{(x^2+3x)^4(2x+3)}dx$.