# Integration by trigonometric substitution

integration by trigonometric substitution:

One of the most powerful techniques of integration is Integration by trigonometric substitution.

Integration by  trigonometric substitution is similar technique to integration by substitution .

In integration by trigonometric substitution we substitute a variable by another trigonometrical variable.

We can integrate the integrals which involves $a^2 - x^2$ , $a^2 + x^2$ or $x^2 - a^2$.

It is obvious that substitution $x = a sin \theta$ turns $a^2 - x^2$ into $a^2 \cos ^2 \theta$ ; the substitution $x = a \tan \theta$ turns $a^2 + x^2$ into into $a^2 \sec ^2 \theta$ and the substitution $x = a \sec \theta$ turns $x^ 2 - a^2$ into  $a^2 \tan ^2 \theta$.

These substitutions makes then resulting function easily integrable.

Examples of integration by trigonometric substitution:

Let us try to integrate the function:

$\int \dfrac{dx}{( a^2 + x^2)^2}$

Let us substitute $x = a \tan \theta$.

Then , $dx = a \sec ^2 \theta . d \theta$

and , $a ^2 + x^2 = a^2 + a^2 \tan ^2 \theta = a^2 \sec ^2 \theta$

It is also quite obvious that  we can the following picture is defined from our substitution:

So,

$\int \dfrac{dx}{( a^2 + x^2)^2} = \int \dfrac{a \sec ^2 \theta . d \theta }{a ^4 \sec ^4 \theta}$

$= \frac{1}{a^3} \int (\cos ^2 \theta .d \theta) = \frac{1}{2 a^3} \int (1+ \cos 2 \theta ) .d \theta$

$= \frac{1}{2 a^3} ( \theta + \frac{1}{2} \sin 2 \theta) + c$

$= \frac{1}{2a^3} ( \theta + \sin \theta . \cos \theta) +c$

$= \frac{1}{2a^3} \left( \tan ^{-1} \frac{x}{a} + \dfrac{ax}{a^2 +x^2} \right) + c$

into  turns Integration by trigonometric substitution

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