∫ tan^2 x sec^3 x dx

If the power of the secant n is odd, and the power of the tangent m is even, then the tangent is expressed as the secant using the identity
1 + tan^2 x = sec^2 x

I thought that since tan is even and sec is odd, we have to put this in terms of cosine and sine.

= ∫ (sin^2 x / cos^2 x) (1/ cos^3 x) dx

= ∫ sin^2 x / cos^5 x dx

u = sin x
du= cos dx

= ∫ sin^2 x cos^-5 x dx

=∫ u^2 cos^-5 x (du/cos x)

= ∫ u^2 / (1 - u^2)^3 du

then split this into two integrals, but how do you do this? Thank you

please for any help. I'm just really confused with this problem.

To split the integral into two integrals, you can use partial fractions decomposition. Here's how you can proceed:

1. Recognize that the integrand can be expressed as u^2 / (1 - u^2)^3. Let's split this into partial fractions.

2. Write the integrand as:

u^2 / (1 - u^2)^3 = A/(1 - u) + B/(1 - u)^2 + C/(1 - u)^3

Here, A, B, and C are constants that we need to find.

3. Multiply both sides of the equation by (1 - u)^3:

u^2 = A(1 - u)^2 + B(1 - u)(1 - u)^2 + C(1 - u)^3

Expanding and simplifying:

u^2 = A(1 - 2u + u^2) + B(1 - u)(1 - 2u + u^2) + C(1 - 3u + 3u^2 - u^3)

u^2 = A(1 - 2u + u^2) + B(1 - 2u + u^2 - u^3) + C(1 - 3u + 3u^2 - u^3)

u^2 = (A + B + C) + (-2A - 2B - 3C)u + (A + B - C)u^2 - Cu^3

4. Equate the coefficients of u^2, u, and the constant term on both sides of the equation:

For the coefficient of u^2: 1 = A + B - C
For the coefficient of u: 0 = -2A - 2B - 3C
For the constant term: 0 = A + B + C

Solving this system of equations will give you the values of A, B, and C.

5. Once you have the values of A, B, and C, you can rewrite the integrand as:

u^2 / (1 - u^2)^3 = A/(1 - u) + B/(1 - u)^2 + C/(1 - u)^3

6. Now, you can integrate each term separately. The integral of A/(1 - u) is Aln|1 - u|, the integral of B/(1 - u)^2 is -B/(1 - u), and the integral of C/(1 - u)^3 is C/(2(1 - u)^2).

7. Finally, substitute back u = sin x into the expressions above and evaluate the integrals.

Note: The above steps provide a general method for splitting the integral into partial fractions and solving for the coefficients. However, the specific values of A, B, and C may require further calculations.