Question

Integrate the following using substitution method or integrate by part ( that suitable the given equation)

Answer 1

Given:

an integral is given as

$\int\frac{y^2+1}{\sqrt{b+y+\frac{1}{3}y^3}}dy$

Find:

we have to evaluate the given integral.

Explanation:

Let us substitute

$\begin{gathered} b+y+\frac{1}{3}y^3=u \\ (1+y^2)dy=du \end{gathered}$

Therefore, given integral becomes

$\int\frac{1}{\sqrt{u}}du=\int u^{-\frac{1}{2}}du=\frac{u^{\frac{1}{2}}}{\frac{1}{2}}+c=2\sqrt{u}+c$

Now, by back substitution, we have

$\int\frac{y^2+1}{\sqrt{b+y+\frac{1}{3}y^3}}dy=2\sqrt{b+y+\frac{1}{3}y^3}+c$

Therefore, the value of the given integral is

$\int\frac{y^2+1}{\sqrt{b+y+\frac{1}{3}y^3}}dy=2\sqrt{b+y+\frac{1}{3}y^3}+c$