Calculus Volume 2 · free preview
Trigonometric Integrals
Integrals of products and powers of sine and cosine reduce through substitution to even powers handled by the half-angle identities, and tangent-secant products reduce by the secant-tangent derivative relation.
1. Sine-cosine products
Sine-cosine products
2. Tangent-secant products
Tangent-secant products
3. The reduction philosophy
The reduction philosophy
4. The two strategies unified
The two strategies unified
5. Odd power strategy
Which strategy integrates sin^3 x cos^2 x dx most directly?
6. Both even strategy
How does one integrate sin^2 x cos^2 x dx, where both exponents are even?
7. Secant cubed
The integral of sec^3 x dx is a classic requiring integration by parts. What is its value?
8. Tangent odd power
For the integral of tan^3 x sec^3 x dx, which substitution applies?
9. Which identity converts
In the even-power strategy for sin^2 x cos^2 x dx, which identity does the conversion?
10. Sin squared integral
The integral of sin^2 x dx equals x/2 minus what fraction of sin 2x? Answer like 1/4.
11. Integral of sec x
The integral of sec x dx equals the logarithm of which expression?
12. Sin times cos
The integral of sin x cos x dx equals one half of sin squared x plus C. As a fraction, what coefficient multiplies sin squared x? Answer like 1/2.
13. Wallis value
Over [0, 2 pi], the integral of sin^2 x dx equals what multiple of pi? Answer with a number.
14. Odd vanishing
Over [0, 2 pi], the integral of sin^3 x dx equals what number, by symmetry? Answer with a number.
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