can you solve this question
Answers
Answer:
( x - arctan x ) / √( 1 + x² )
Step-by-step explanation:
The arctan(x) and the 1+x² are both very suggestive of making the substitution x = tan u. So let's do that.
Put x = tan u. Then 1+x² = sec² u and dx = sec² u du.
Hence ( 1 + x² )^(3/2) = sec³ u so
dx / ( 1 + x²)^(3/2) = du / sec u = cos u du.
Now the integral becomes
∫ (tan u) (u) cos u du
= ∫ u sin u du
= sin u - u cos u (this step is a basic integration by parts) (***)
Going back to x = tan u, we saw 1 + x² = sec² u, so solving for cos u we get
cos u = 1 / √( 1 + x² )
and using sin² u = 1 - cos² u, we also get
sin u = x / √( 1 + x² ).
Putting this into (***) to get the answer in terms of x, we finish with the integral being
( x - arctan x ) / √( 1 + x² )
Answer:
Step-by-step explanation:
