How do I find the indefinite integral of: sec(y)(tan(y) - sec(y)) dy
To find the indefinite integral of the given function, sec(y)(tan(y) - sec(y)) dy, we can use the method of substitution.
Step 1: Start by letting u = tan(y). Then, du = sec^2(y) dy.
Rearranging the equation, we have dy = du / sec^2(y).
Step 2: Substitute these expressions into the integral:
∫ sec(y)(tan(y) - sec(y)) dy = ∫ sec(y)(u - sec(y)) (du / sec^2(y))
Step 3: Simplify the expression:
∫ (u - sec^2(y)) du.
Step 4: Distribute sec(y) into the expression:
∫ u du - ∫ sec^2(y) du.
Step 5: Integrate each term:
The integral of u du is (u^2 / 2) + C.
The integral of sec^2(y) du is tan(y) + C.
Step 6: Putting it all together, we get:
∫ sec(y)(tan(y) - sec(y)) dy = (u^2 / 2) - tan(y) + C.
Step 7: Finally, we can substitute u back in terms of y:
(tan^2(y) / 2) - tan(y) + C.
Therefore, the indefinite integral of sec(y)(tan(y) - sec(y)) dy is (tan^2(y) / 2) - tan(y) + C, where C is the constant of integration.
no sweat. Just expand the integrand and you have
sec(y) tan(y) - sec^2(y)
Those look familiar?