Integration by Substitution

Some functions may be changed to the standard forms by easy mathematical manipulation.
To aid the process of changing to a recognisable form, use it also made of substitution, which results in a change of variable. In mathematics, an important use is made of what is called differentials.
$$ \begin{align} \displaystyle
u &= f(x) \\
du &= \dfrac{du}{dx} \times dx \\

Example 1

Find $\displaystyle 2\int{\sqrt{2x+1}}dx$.

Example 2

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

Example 3

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

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