To determine

To evaluate:

The given integral.

Answer

arctan⁡x33+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:

∫arctan⁡x2x2+1 dx

4) Calculation:

Consider the given integral, ∫arctan⁡x2x2+1 dx

Here, using the substitution method

Substitute arctan⁡x=u,

Differentiating with respect to x

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

∫arctan⁡x2x2+1 dx= ∫u2du

By using the power rule of indefinite integral

=u33+c

Resubstituting u=arctan⁡x in the above solution

=arctan⁡x33+c

Conclusion:

Therefore, ∫arctan⁡x2x2+1 dx= arctan⁡x33+c