∫sinⁿ(x)cosᵐ(x)dx | Solve all Four Potential Scenarios

Walkthrough all processes for integrals of the form: ∫sinⁿ(𝒙)cosᵐ(𝒙)𝒅𝒙.

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0:00 Integrate sinⁿ(𝒙)cosᵐ(𝒙)𝒅𝒙
0:12 Four odd and even power cases
0:28 Case 1: Sine odd, cosine even
0:38 Rewriting sin^5(x)cos^2(x)
1:25 Using u = cos(x)
2:13 Expanding and integrating the polynomial
2:52 Final answer for sin^5(x)cos^2(x)
3:08 Key idea for odd sine powers
3:26 Case 2: Sine even, cosine odd
3:37 Rewriting sin^4(x)cos^3(x)
4:12 Using u = sin(x)
4:43 Integrating the second example
5:04 Key idea for odd cosine powers
5:17 Case 3: Both powers odd
5:36 Choosing which trig factor to save
5:58 Using u = sin(x)
6:20 Final answer for sin^3(x)cos^3(x)
6:44 Case 4: Both powers even
7:03 Power reduction identities
7:18 Example: sin^2(x)cos^2(x)
7:33 Applying power reduction
8:03 Difference of squares step
8:25 Reducing cos^2(2x)
8:47 Simplifying to one integral
9:13 Integrating 1 - cos(4x)
10:03 Final answer for sin^2(x)cos^2(x)
10:21 Key idea for even powers
10:29 Summary of all four cases
11:20 Thanks for watching!

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