Integrand 

Substitution

$\sqrt[n]{g\left(x\right)}$

$u\=\sqrt[n]{g\left(x\right)}$ or $u\=g\left(x\right)$ or $u^n\=g\left(x\right)$

$\sqrt[n]{g\left(x\right)}$ and $\sqrt[m]{g\left(x\right)}$ 

$u\=\sqrt[k]{g\left(x\right)}$, $k$ = least common multiple of $m$ and $n$ 

Rational function of $\sin \left(x\right)$ and $\cos \left(x\right)$ 

$z\=\tan \left(\frac{x}{2}\right)\,\mathrm{dx}\=\frac{2 \mathrm{dz}}{1\+z^2}\,\sin \left(x\right)\=\frac{2 z}{1\+z^2}\,\cos \left(x\right)\=\frac{1-z^2}{1\+z^2}$

Table 6.6.1   Common rationalizing substitutions

From $z\=\tan \left(x\/2\right)$ and Figure 6.6.1 it follows that

$\sin \left(\frac{x}{2}\right)\=\frac{z}{\sqrt{1\+ z^2}}$ and $\cos \left(\frac{x}{2}\right)\=\frac{1}{\sqrt{1\+ z^2}}$ 

From the trig identities

$\sin \left(x\right)\=2 \sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)$ 

and

$\cos \left(x\right)\={\cos }^2\left(\frac{x}{2}\right)-{\sin }^2\left(\frac{x}{2}\right)$ 

it follows that

$\sin \left(x\right)\=\frac{2 z}{1\+ z^2}$ and $\cos \left(x\right)\=\frac{1-z^2}{1\+ z^2}$ 

Since $x\=2 \arctan \left(z\right)$, then $\mathrm{dx}\=\frac{2 \mathrm{dt}}{1\+z^2}$

Example 6.6.1

Evaluate the indefinite integral $\int \frac{x}{\sqrt{x-1}} \mathit{\ⅆ}x$.

Example 6.6.2

Evaluate the indefinite integral $\int \frac{1}{x^{1\/3}\+x^{1\/4}} \mathit{\ⅆ}x$.

Example 6.6.3

Evaluate the indefinite integral $\int \frac{1}{2 \sin \left(x\right)\+3 \cos \left(x\right)} \mathit{\ⅆ}x$.

Example 6.6.4

Evaluate the integral $\int \frac{1}{2 \sin \left(x\right)\+3 \cos \left(x\right)} \mathit{\ⅆ}x$ without making the rationalizing substitution $z\=\tan \left(x\/2\right)$.

Example 6.6.5

Obtain a continuous antiderivative for $1\/\left(7\+2 \sin \left(x\right)\+3 \cos \left(x\right)\right)$.