Concept:

• \(\rm \int\frac{1}{x+a}dx=log|x+a|+C\)
• \(\rm \int\frac{1}{(x+a)^n}dx=\frac{1}{(1-n)(x+a)^{n+1}}+C\)

Calculation:​

To solve: \(\rm \int \frac{2x^3}{(x^2+7)^2}dx\) 

Let us put x² = t ⇒2xdx = dt in \(\rm \int \frac{2x^3}{(x^2+7)^2}dx\)  

⇒ \(\rm \int \frac{2x^3}{(x^2+7)^2}dx = \rm \int \frac{t}{(t+7)^2}dt\)

This integrand is a proper rational fraction. therefore, by using the form of partial fraction, we write

⇒ \(\rm \frac{t}{(t+7)^2}=\frac{A}{t+7}+\frac{B}{(t+7)^2}\)

⇒ t = At + 7A + B 

By comparing coefficient of t and constant terms on both sides, we get A = 1 and 7A + B = 0

By solving these equation, we get  A = 1 and B = -7

\(⇒ \rm \frac{t}{(t+7)^2}=\frac{1}{t+7}-\frac{7}{(t+7)^2}\)

⇒ \(\rm \int\frac{t}{(t+7)^2}dt=\int\frac{1}{t+7}dt-\int\frac{7}{(t+7)^2}dt\)

⇒ \(\rm \int\frac{t}{(t+7)^2}dt=log\left | t+7 \right |+\frac{7}{(t+7)}+C\) 

Now put t = x² in the above equation we get

⇒ \(\rm \int\frac{2x^3}{(x^2+7)^2}dx=log\left | x^2+7 \right |+\frac{7}{(x^2+7)}+C\)