Integration by Substitution/Examples

Examples of Use of Integration by Substitution

Primitive of $\paren {2 x + 3} \sqrt {x^2 + 3 x + 2}$

$\ds \int \paren {2 x + 3} \sqrt {x^2 + 3 x + 2} \rd x = \dfrac 2 3 {\paren {\sqrt {x^2 + 3 x + 2} }^3} + C$

Primitive of $\frac {\cos x} {\paren {1 + \sin x}^2}$

$\ds \int \dfrac {\cos x} {\paren {1 + \sin x}^2} \rd x = -\dfrac 1 {1 + \sin x} + C$

Primitive of $\sqrt {1 - x}$

$\ds \int \sqrt {1 - x} \rd x = -\dfrac 2 3 \paren {1 - x}^{3 / 2} + C$