What is the integral of sin(x)/cos^4 (x)

Question

Please help me answer this, thank you

Answer 1

∫sin(x)/cos^4(x) dx

Use the substitution dx=1/t • dt where t=cos(x) and t'=-sin(x)

∫sin(x)/cos^4(x) •1/-sin(x) dt

sin(x) cancels out and you get:

∫-1/cos^4(x) dt

Now substitute t for cos(x):

∫-1/t^4 dt

Evaluate the integral:

-(-1/3t^3)

Substitute back t=cos(x)

-(-1/3(cos^3(x)))

Simplify the expression:

1/(3cos^3(x))

Add the constant of integration:

1/(3cos^3(x)) + C

I hope that helps! :)

Answer 2

If you rewrite this as sec²x sec x tan x, the answer is obvious: 1/3 sec³ x + C or 1/(3cos³x) + C