Some Basic Substitutions of Integration

In order to evaluate some particular integrals, some particular substitutions play very important role in making the integral easier to evaluate it. Now we have some basic substitutions of integrals as follows:

Expression Involving
Suitable Substitution
\sqrt {{a^2} - {x^2}}
x = a\sin \theta or x = a\cos \theta
\sqrt {{x^2} - {a^2}}
x = a\sec \theta or x = a\cosh \theta
\sqrt {{a^2} + {x^2}}
x = a\tan \theta or x = a\sinh \theta
\sqrt {x - a} or \sqrt {x + a}
\sqrt {x - a} = t or \sqrt {x + a} = t
\sqrt {2ax - {x^2}}
x - a = a\sin \theta or x - a = a\cos \theta
\sqrt {2ax + {x^2}}
x - a = a\sec \theta or x - a = a\cosh \theta