Calc help?

Question

Integration:

1) 1/(x^2+4x+5) dx
2) e^x/(e^x+1)dx
3) e^x+1/e^xdx
4) x square root (ax+b)

Thanks

Answer 1

1) arctan (x+2)
2) ln (e^x + 1)
3) 2 sinh (x)

On number 4, is it dx? That one is really big...

4) (2*(ax+b)^(3/2)*(3ax-2b)) / (15a^2)

Answer 2

1: ∫1/(x²+4x+5) dx
∫1/(x²+4x+4 + 1) dx
∫1/((x+2)²+1) dx
Let tan u = x+2, dx=sec² u du
∫1/(tan² u + 1) sec² u du
∫1/sec² u sec² u du
∫1 du
u+C
arctan ((x+2)²) + C

2: ∫e^x/(e^x+1) dx
u=e^x+1, du=e^x dx
∫1/u du
ln u + C
ln (e^x+1) + C

3: ∫e^x+1/e^x dx
∫e^x dx + ∫1/e^x dx
e^x + ∫e^(-x) dx
e^x - e^(-x) + C

4: ∫x√(ax+b) dx
u=ax+b, du=a dx, dx=du/a, x=(u-b)/a
∫1/a x√u du
∫(u-b)/a² √u du
1/a² (∫u√u du - b∫√u du)
(∫u^(3/2) du - b∫u^(1/2) du)/a²
2u^(5/2)/(5a²) - 2bu^(3/2)/(3a²) + C
2(ax+b)^(5/2)/(5a²) - 2b(ax+b)^(3/2)/(3a²) + C

Answer 3

4. X2/2*sqrt(ax+b)^1.5/3a

1. (X2+4x+5)^-1=9(X2+4X+5)^0/(2X+4)(-1))=-(1/2X+4)