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+(1)/(3)y^3}}dy`

Find:

we have to evaluate the given integral.

Step-by-step explanation:

Let us substitute

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

Therefore, given integral becomes

`\int(1)/(√(u))du=\int u^{-(1)/(2)}du=\frac{u^{(1)/(2)}}{(1)/(2)}+c=2√(u)+c`

Now, by back substitution, we have

`\int\frac{y^2+1}{\sqrt{b+y+(1)/(3)y^3}}dy=2\sqrt{b+y+(1)/(3)y^3}+c`

Therefore, the value of the given integral is

`\int\frac{y^2+1}{\sqrt{b+y+(1)/(3)y^3}}dy=2\sqrt{b+y+(1)/(3)y^3}+c`