Some Basic Substitutions of Integration

In order to evaluate some particular integrals, some substitutions play very important roles in making the integral easier to evaluate. We have some basic substitutions of integrals, which are as follows:

Expression Involved
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