Calculus II ep15: Trigonometric integrals (Oct 14, 2026)

Overview

Prof. Staecker presents two methods for integrating products of powers of sine and cosine: save one factor from a trig function with an odd exponent and use a substitution, or use half-angle identities when all exponents are even. Worked examples show how to convert the integrands into polynomials or sums of cosines, then integrate term by term while tracking substitution signs and frequency constants.

Key takeaways

For a product of sine and cosine powers with at least one odd exponent, save one factor from an odd-powered function, rewrite the rest using sin²(x) + cos²(x) = 1, and substitute the other trig function.
When both exponents are odd, either function can be chosen for the saved factor, but saving sine versus cosine produces answers in different-looking forms.
If every exponent is even, convert squared factors with sin²(x) = [1 − cos(2x)]/2 and cos²(x) = [1 + cos(2x)]/2 instead of using the save-one method.
Reducing cos²(2x) or sin²(2x) introduces a fourth-angle term, such as cos²(2x) = [1 + cos(4x)]/2, which can then be integrated directly.
For ∫cos(kx) dx, divide by the inside frequency k; omitting this factor gives an incorrect antiderivative.
In the save-one method, a substitution such as u = cos(x) gives du = −sin(x) dx, so the negative sign must remain in the transformed integral.

Chapters

0:00 When Trigonometric Integrals Need Specialized Tricks

The lesson focuses on products of powers such as cos⁴(x) sin³(x), rather than general-purpose integration methods.
The first method relies on sin²(x) + cos²(x) = 1, rewritten as cos²(x) = 1 − sin²(x) or sin²(x) = 1 − cos²(x).

2:40 Save One Cosine and Substitute u = sin(x)

For ∫cos³(x) dx, separate one cosine factor and rewrite the other two as cos²(x) = 1 − sin²(x).
Set u = sin(x), so du = cos(x) dx; the integral becomes ∫(1 − u²) du.
Integrating the polynomial and substituting back gives sin(x) − sin³(x)/3 + C.

7:30 Use the Odd Exponent in a Mixed-Power Integral

For an integrand such as sin²(x)cos⁵(x), save one cosine because its exponent is odd.
Rewrite cos⁴(x) as (1 − sin²(x))², then use u = sin(x) and du = cos(x) dx.
Expand the resulting polynomial in u and integrate each power; saving an odd-powered factor makes the substitution work.

13:00 Practice the Save-One Method with sin³(x)cos³(x)

Both exponents are odd, so either a sine or cosine factor can be saved; the class saves one sine to match answers.
Rewrite the remaining sin²(x) as 1 − cos²(x), then set u = cos(x).
Because du = −sin(x) dx, carry a negative sign through the substitution before integrating the polynomial.

20:00 Choose Half-Angle Identities When All Powers Are Even

The save-one method applies when at least one sine or cosine exponent is odd; two even exponents call for a different method.
Use sin²(x) = [1 − cos(2x)]/2 and cos²(x) = [1 + cos(2x)]/2 to reduce squared trig factors.
For squared double-angle terms, substitute 2x into the identities; for example, cos²(2x) = [1 + cos(4x)]/2.
Integrating cos(kx) requires the frequency factor: ∫cos(kx) dx = sin(kx)/k + C.

26:30 Reduce sin⁴(x) to Integrable Cosine Terms

Write sin⁴(x) as [sin²(x)]² and replace sin²(x) with [1 − cos(2x)]/2.
Squaring and expanding produces a cos²(2x) term, which reduces using cos²(2x) = [1 + cos(4x)]/2.
The resulting constant, cos(2x), and cos(4x) terms can be integrated separately using the frequency rule.

32:00 Apply Both Half-Angle Identities to sin²(x)cos²(x)

Replace sin²(x) and cos²(x) with [1 − cos(2x)]/2 and [1 + cos(2x)]/2, respectively.
The product simplifies through (1 − a)(1 + a) = 1 − a², with a = cos(2x).
Replace cos²(2x) with [1 + cos(4x)]/2, leaving a constant and a cos(4x) term to integrate.

35:10 Practice the Even-Power Method on cos⁴(x)

Rewrite cos⁴(x) as [cos²(x)]² and substitute cos²(x) = [1 + cos(2x)]/2.
Expand the square, then reduce cos²(2x) using [1 + cos(4x)]/2.
Combine the constant terms and integrate the cosine terms, remembering to divide by 2 for cos(2x) and by 4 for cos(4x).

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Prof Staecker.