If the integrand contains a single factor of one of the forms we can try a trigonometric substitution.
 If the integrand contains let x = asinθ and use the identity 1 − sin^{2}θ = cos^{2}θ.
 If the integrand contains let x = atanθ and use the identity 1 + tan^{2}θ = sec^{2}θ.
 If the integrand contains let x = asecθ and use the identity sec^{2}θ − 1 = tan^{2}θ.
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If the integrand contains a piece of the form we use the substitution
This will transform the integrand to a trigonometic function. If the new integrand can't be integrated on sight then the tanhalfangle substitution described below will generally transform it into a more tractable algebraic integrand.
Eg, if the integrand is √(1x^{2}),
If the integrand is √(1+x)/√(1x), we can rewrite it as
Then we can make the substitution
When the integrand contains a piece of the form we use the substitution
E.g, if the integrand is (x^{2}+a^{2})^{3/2} then on making this substitution we find
If the integral is
then on making this substitution we find
After integrating by parts, and using trigonometric identities, we've ended up with an expression involving the original integral. In cases like this we must now rearrange the equation so that the original integral is on one side only
As we would expect from the integrand, this is approximately z^{2}/2 for large z.
If the integrand contains a factor of the form we use the substitution
Find
Find
We can now integrate by parts
