Let f be a differentiable function with the following values:

Question

Let f be a differentiable function with the following values:

x:     0 1

f(x): 6 10

f'(x):4 11

a) ∫0 to 1 of f'(x)(sin(f(x)))dx

Using substitution I got the integral to be

∫6 to 10 sin(u)

where I took u to be f(x). I am a little confused on the next step of how to evaluate the integral to get the answer. I tried 11sin(10)-4sin(6) but I was marked wrong.

Same for part b)

it asks ∫0 to pi/2 sinx(f'(cosx))dx

I found this integral to be ∫0 to 1 f'(u)du but I don't know what to do next. Please help me, because on this one I don't even have a guess on how to evaluate it.

Answer 1

Given:  f(0) = 6    f(1) = 10

 

            f'(0) = 4   f'(1) = 11

 

Let u = f(x), then du = f'(x)dx   When x = 0, u = f(0) = 6 and when x = 1, u = f(1) = 10

 

a. So, ∫_{from 0 to 1} sin(f(x) f'(x)dx = ∫_{from 6 to 10} sinudu = -cosu _{from 6 to 10}

 

                                                                             = -(cos10 - cos6)

 

                                                                             = cos6 - cos10

 

b. Let u = cosx.  Then du = -sinxdx.  So, sinxdx = -du

 

    When x = 0, u = cos0 = 1  and when x = π/2, u = cos(π/2) = 0

 

    ∫_{from 0 to π/2} sinxf'(cosx)dx = ∫_{from 1 to 0} f'(u)(-du)

 

                                            = ∫_{from 0 to 1}f'(u)du = f(u)_{from 0 to 1}

 

                                                                         = f(1) - f(0) = 10 - 6 = 4