Evaluate the indefinite integral.
∫ sec^3x tanx dx
I let u=tanx
took derivative..
du=sec^2x dx
..now what??
sec^3 x tan x dx
| = integral symbol
Rewrite as
| sec^2 x sec x tan x dx
u = sec x
du = sec x tan x
| u^2 du
1/3 u^3 + C
Substitute back in, u = sec x
1/3 sec^3 x + C
Now that you have found the derivative of u with respect to x, you can substitute it into the integral:
∫ sec^3x tanx dx = ∫ u du
Next, you can solve the indefinite integral of u with respect to u:
∫ u du = u^2/2 + C
Finally, substitute u back in:
∫ sec^3x tanx dx = tan^2x/2 + C, where C is the constant of integration.