Ronnie Baur

2021-11-19

Find the indefinite integral.

$\int x{({x}^{2}+1)}^{2}dx$

Oung1985

Beginner2021-11-20Added 16 answers

Step 1

We have to find the indefinite integral:

$\int x{({x}^{2}+1)}^{2}dx$

We will solve this integrals using substitution method:

Let$t={x}^{2}+1$

Differentiating with respect to x, we get

$t={x}^{2}+1$

dt=2xdx+0

$\frac{dt}{2}=xdx$

So putting the value of xdx in the given indefinite integration, we get

Step 2

Here,

$\int x{({x}^{2}+1)}^{2}dx=\int {({x}^{2}+1)}^{2}xdx$

$=\int {t}^{2}\left(\frac{dt}{2}\right)$

$=\frac{1}{2}\int {t}^{2}dt$

$=\frac{1}{2}\left(\frac{{t}^{2+1}}{2+1}\right)+c$

$=\frac{{t}^{3}}{6}+c$ .

(using formula$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+c$ )

Where c is an arbitrary constant.

Putting$t={x}^{2}+1$ , we get

$=\frac{{({x}^{2}+1)}^{3}}{6}+c$ .

Hence, value of indefinite integration is$\frac{{({x}^{2}+1)}^{3}}{6}+c$ .

We have to find the indefinite integral:

We will solve this integrals using substitution method:

Let

Differentiating with respect to x, we get

dt=2xdx+0

So putting the value of xdx in the given indefinite integration, we get

Step 2

Here,

(using formula

Where c is an arbitrary constant.

Putting

Hence, value of indefinite integration is

Daniel Williams

Beginner2021-11-21Added 14 answers

Explanation:

Expand

Distribute the x

Next we integrate each term

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