Evaluate the following integral. integrate of 0 to (pi/4) of cos(x)sin(sin(x))dx

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You should use the following substitution to solve the given integral such that:

`sin x = t => cos x dx = dt`

Changing the variable yields:

`int cos x*sin(sin x)dx = int sin t dt`

`int sin t dt = -cos t + c`

Substituting back `sin x`  for `t`  yields:

`int cos x*sin(sin x)dx = - cos(sin x) + c`

You may evaluate the definite integral using fundamental theoem of calculus such that:

`int_(x_1)^(x_2) f(x) dx = F(x_2) - F(x_1)`

Reasoning by analogy yields:

`int_0^(pi/4) cos x*sin(sin x)dx = - cos(sin x)|_0^(pi/4)`

`int_0^(pi/4) cos x*sin(sin x)dx = - cos(sin (pi/4)) + cos(sin 0)`

`int_0^(pi/4) cos x*sin(sin x)dx = - cos(sqrt2/2) + cos 0`

Hence, evaluating the given definite integral yields `int_0^(pi/4) cos x*sin(sin x)dx = cos 0 - cos(sqrt2/2). `

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial