Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 14/Important Transformations

Important Transformations

Often in practice an integral can be simplified by using an appropriate transformation or substitution and formula $14.6$. The following list gives some transformations and their effects.

$14.49$: Primitive of $F \left({a x + b}\right)$

$14.50$: Primitive of $F \left({ \sqrt{a x + b} }\right)$

$14.51$: Primitive of $F \left({ \sqrt [n] {a x + b} }\right)$

$14.52$: Primitive of $F \left({ \sqrt{a^2 - x^2} }\right)$

$14.53$: Primitive of $F \left({ \sqrt{x^2 + a^2} }\right)$

$14.54$: Primitive of $F \left({ \sqrt{x^2 - a^2} }\right)$

$14.55$: Primitive of $F \left({e^{a x}}\right)$

$14.56$: Primitive of $F \left({\ln x}\right)$

$14.57$: Primitive of $F \left({\sin^{-1} \dfrac x a}\right)$

Similar results apply for other inverse trigonometric functions:

Primitive of $F \left({\cos^{-1} \dfrac x a}\right)$

Primitive of $F \left({\tan^{-1} \dfrac x a}\right)$

Primitive of $F \left({\cot^{-1} \dfrac x a}\right)$

Primitive of $F \left({\sec^{-1} \dfrac x a}\right)$

Primitive of $F \left({\csc^{-1} \dfrac x a}\right)$

$14.58$: Primitive of $F \left({\sin x, \cos x}\right)$