COMMON SUBSTITUTIONS

1.

$\displaystyle \int F(ax+b)dx = \displaystyle \frac{1}{a} \displaystyle \int F(u)du$
where $u=ax\,+\,b$

2.

$\displaystyle \int F\left(\displaystyle\sqrt{ax\, +\, b}\right)\, dx = \displaystyle \frac{2}{a} \displaystyle \int u\,F(u)\,du$
where $u=\displaystyle\sqrt{ax\,+\,b}$

3.

$\displaystyle \int F\left( \sqrt[n]{ax+b} \right) \,dx = \displaystyle \frac{n}{a} \displaystyle \int u^{n-1}\,F(u)\,du$
where $u=\sqrt[n]{ax+b}$

4.

$\displaystyle \int F\left( \displaystyle\sqrt{a^{2}-x^{2}}\right)\,dx = a\,\displaystyle \int F(a \cos u)\,\cos u\,du$
where $x=a\sin u$

5.

$\displaystyle \int F\left( \displaystyle\sqrt{x^2+a^{2}} \right)\,dx= a\,\displaystyle \int F(a \sec u) \sec ^{2} u \, du$
where $x=a\tan u$

6.

$\displaystyle \int F\left( \displaystyle\sqrt{x^{2}-a^{2}} \right)\,dx=a \displaystyle \int F(a\tan u) \sec u \tan u\,du$
where $x=a\sec u$

7.

$\displaystyle \int F (e\displaystyle^{ax})\,dx = \displaystyle \frac{1}{a} \displaystyle \int\displaystyle \frac{F(u)}{u}\,du$
where $u=e\displaystyle^{ax}$

8.

$\displaystyle \int F(\ln x)\,dx = \displaystyle \int F(u)\,e\displaystyle^u\,du$
where $u=\ln x$

9.

$\displaystyle \int F\left( \sin ^{-1}\displaystyle \frac{x}{a}\right)\,dx = a\,\displaystyle \int F(u)\cos u\,du$
where $u=\sin ^{-1}\displaystyle \frac{x}{a}$