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