Question

Find the value of SÓ x? (1 - x)”dx​

Answer 1

The best way to solve this kind of integral is by u-substitution of  1+x  and using the sum rule and power rule for the rest.

To demonstrate:

∫x1+x−−−−−√dx  

Substitute  u=1+x . compute  dudx(1+x)=1 , so  du=dx . Since we can express  x=x+1–1 , so we have  ∫(u−1)u−−√du  

We can express  u−−√=u12 , so:  ∫(u−1)u12du  

Distribute the parenthesis:  =∫u12u−u12du⇒∫u32−u12du  

Use the sum rule:  ∫u32du−∫u12du  

So now we solve the integral term by term by using the power rule for each term.

Solving for  ∫u32du=u32+132+1⇒2u525  

Solving for  ∫u12du=u12+112+1⇒2u323  

Plug in solved integrals:  2u525−2u323  

Undo substitution of  u=x+1 :  =2(x+1)525−2(x+1)323  

And that is all there is to it. Add the constant of integration to finish the solution:

=2(x+1)525−2(x+1)323+C

Answer 2

Step-by-step explanation:

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