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

Expert Answers
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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). `