Question

Use the substitution method to evaluate the integral
integrate dy/2(y^1/2)(1+y^1/2 )^2) dx, x=1..4

Answer 1

its easy jus write the equation using equations, im having trouble reading it

Answer 2

ohh i got it

Answer 3

cool

Answer 4

962/15

Answer 5

What's that Aron? I need steps if you can perform the intermediate steps

Answer 6

Possible intermediate steps:
integral 2 (1+sqrt(y))^2 sqrt(y) dy
Factor out constants:
= 2 integral (sqrt(y)+1)^2 sqrt(y) dy
For the integrand (sqrt(y)+1)^2 sqrt(y), substitute u = sqrt(y) and du = 1/(2 sqrt(y)) dy:
= 4 integral u^2 (u+1)^2 du
Expanding the integrand u^2 (u+1)^2 gives u^4+2 u^3+u^2:
= 4 integral (u^4+2 u^3+u^2) du
Integrate the sum term by term and factor out constants:
= 4 integral u^4 du+8 integral u^3 du+4 integral u^2 du
The integral of u^2 is u^3/3:
= 4 integral u^4 du+(4 u^3)/3+8 integral u^3 du
The integral of u^3 is u^4/4:
= 2 u^4+4 integral u^4 du+(4 u^3)/3
The integral of u^4 is u^5/5:
= (4 u^5)/5+2 u^4+(4 u^3)/3+constant
Substitute back for u = sqrt(y):
= (4 y^(5/2))/5+(4 y^(3/2))/3+2 y^2+constant

Now just substitute the required values!