Indefinite Integral?

Question

Evaluate the indefinite integral:

⌠ cosx/ 5sinx +25
⌡

Answer 1

Is the denominator 5 sin x or 5 sin x + 25?
If it's 5 sin x, then the integral is
1/5∫ cot x + 25 = 1/5(ln|sin x| ) + 25 x + C.
If it's 5 sin x + 25, we get
1/5 ∫ cos x/(sin x + 5)
Let u = sin x, du = cos x dx.
Then we have
1/5 ∫ du/(u+5)= 1/5 ln| u+5| = 1/5 ln|sin x + 5| + C.

Answer 2

First, break it up into int(cos(x)/5sin(x)) + int(25).
Let u=sin(x)
du=cos(x)dx
Solving for dx
du/cos(x)=dx
Substitute into your first integral
int(cos(x)/5sin(x) * du/cos(x))
cos cancels and we said that u=sin(x) so
int(1/5u du)
Bring out the 1/5
1/5int(1/u du)
Integrate
1/5ln(u)
Change your variable back to x
1/5ln(sin(x))
Do your second integral
int(25)=25x
Put it all together
1/5ln(sin(x))+25x
Don't forget your constant of integration!
1/5ln(sin(x))+25x+C