Tutorial Exercise

Find the indefinite integral by using the substitution $x=7tan().$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{7x^2}{(49+x^2{)}^2}dx$

Step $1$

To solve the indefinite integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{7x^2}{(49+x^2{)}^2}dx,$ rewrite the integrand as follows.

$\frac{7x^2}{(49+x^2{)}^2}=\frac{7x^2}{(\sqrt[\phantom{2}]{(49+x^2)}{)}^4}$

Step $2$

Use the trigonometric substitution $x=7tan().$ Differentiate with respect to $$ on both sides of expression.

$dx=$

$$

Rewrite the integral by substituting for $x.$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{7x^2}{\sqrt[\phantom{2}]{(49+x^2{)}^4}}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(\frac{7(7tan(){)}^2}{(x{)}^4})$