∫sin^5 t/cos^2 t dt

To find the integral ∫sin^5(t)/cos^2(t) dt, we can use the u-substitution method. Let's break down the steps:

Step 1: Identify the appropriate substitution.
In this case, let's substitute u = cos(t) because it appears in the denominator.

Step 2: Find the derivative.
To find du, differentiate both sides of the substitution equation u = cos(t) with respect to t.
du/dt = -sin(t)

Step 3: Solve for dt.
Rearrange the previous equation to solve for dt:
dt = du / (-sin(t))

Step 4: Replace dt and sin(t) in the integral.
Now, we substitute the expression for dt and sin(t) in the integral using the substitution:
∫sin^5(t)/cos^2(t) dt = ∫(sin(t)^5 / cos^2(t)) * (du / -sin(t))

Step 5: Simplify the integral.
Simplify the expression inside the integral by canceling out sin(t):
∫sin(t)^4 / cos^2(t) du

Step 6: Apply trigonometric identity.
Rewrite sin(t)^4 as (1 - cos(t)^2)^2 using the identity sin^2(t) = 1 - cos^2(t):
∫(1 - cos(t)^2)^2 / cos^2(t) du

Step 7: Expand and rationalize the expression.
Expand the numerator and combine like terms:
∫(1 - 2cos(t)^2 + cos(t)^4) / cos^2(t) du

Distribute the denominator into each term:
∫(1/cos^2(t)) - (2cos(t)^2/cos^2(t)) + (cos(t)^4/cos^2(t)) du

Simplify:
∫sec^2(t) - 2 + cos^2(t) du

Step 8: Integrate term by term.
Now we can integrate each term individually:
∫sec^2(t) du - ∫2 du + ∫cos^2(t) du

The integral of sec^2(t) is tan(t), the integral of 2 is 2u, and the integral of cos^2(t) can be computed using the double-angle formula for cosine:
∫sec^2(t) du - ∫2 du + ∫(1/2)(1 + cos(2t)) du

Step 9: Simplify and combine.
Simplifying each term and combining them, we get:
tan(t) - 2u + (1/2)(u + (1/2)sin(2t)) + C

Step 10: Substitute back.
Finally, substitute back u = cos(t) to get the final result:
tan(t) - 2cos(t) + (1/2)(cos(t) + (1/2)sin(2t)) + C

Therefore, the integral ∫sin^5(t)/cos^2(t) dt is equal to tan(t) - 2cos(t) + (1/2)(cos(t) + (1/2)sin(2t)) + C, where C is the constant of integration.