Help me solve these integration questions?

Question

1) integral x/(x+1)^1/2 dx
2) integral x^3(x^2+1)^1/2 dx
Please show the workings clearly, thanks

Answer 1

1. ∫x / (x+1)^(1/2) dx; let u = x+1, du = dx
= ∫(u-1).u^(-1/2) du
= ∫(u^(1/2) - u^(-1/2)) du
= (u^(3/2) / (3/2)) - (u^(1/2) / (1/2)) + c
= (2/3) (x+1)^(3/2) - 2 (x+1)^(1/2) + c.

2. ∫x^3 (x^2+1)^(1/2) dx; let u = x^2 + 1, du = 2x dx
= ∫(u-1) u^(1/2) (1/2) du
= (1/2) ∫(u^(3/2) - u^(1/2)) du
= (1/2) [u^(5/2) / (5/2) - u^(3/2) / (3/2)] + c
= (1/5) (x^2 + 1)^(5/2) - (1/3) (x^2 + 1)^(3/2) + c.

Answer 2

1
put t^2=x+1
differentiatin
2 t dt =dx
intgeral of x/x+1^1/2 dx
= integral((t^2-1)/t)2tdt
= 2 t^3/3-2t
put the value of t =(x+1)^1/2

Answer 3

1)multiply and divide to (x+1)^1/2
2)Convert to "sin" model

Answer 4

lol