#### To determine

**To evaluate:**

The given integral.

#### Answer

arctanx33+c

#### Explanation

**1) Concept:**

The substitution rule: If u=g(x) is a differentiable function whose range is I, and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du. Here g(x) is substituted as u and then g’(x)dx =du

**2) Formula:**

i.

∫cf(x)dx=c∫f(x)dx

ii.

**∫xndx=xn+1n+1**

**3) Given:**

∫arctanx2x2+1 dx

**4) Calculation:**

Consider the given integral, ∫arctanx2x2+1 dx

Here, using the substitution method

Substitute arctanx=u,

Differentiating with respect to x

11+x2 dx=du Using this in the given integral, it becomes

∫arctanx2x2+1 dx= ∫u2du

By using the power rule of indefinite integral

=u33+c

Resubstituting u=arctanx in the above solution

=arctanx33+c

**Conclusion:**

Therefore, ∫arctanx2x2+1 dx= arctanx33+c