COMMON SUBSTITUTIONS

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

8.

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

9.

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

[Tables]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Copyright © 1999-2024 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour