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:

Integration by trigonometric substiotution

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


Related posts:

  1. Integration by Substitution Integration by substitution , how to integrate a integral by...
  2. Derivatives of inverse trigonometric functions Inverse trigonometric functions  are the  inverse of trigonometric functions ....
  3. Integration Formulas Using Integration formulas is one of the most basic and...
  4. Derivatives of Trigonometric functions. As you know, The functions SINE x(sin x) , CO-SECANT...
  5. Techniques of Integration Main techniques of finding integration of a function. Techniques of...