﻿ ﻿TRIGONOMETRIC INTEGRALS - Special Methods of Integration - The Calculus Primer

## The Calculus Primer (2011)

### Chapter 48. TRIGONOMETRIC INTEGRALS

13—3. Trigonometric Reduction. Many differentials containing trigonometric functions can be reduced to standard forms for integration by first making appropriate trigonometric transformations.

EXAMPLE 1. Find sin2 x dx.

Solution. By trigonometry, sin2

NOTE: This may also be written, by trigonometry, as

since sin 2x = 2 sin x cos x.

EXAMPLE 2. Find sin3 θ dθ.

Solution. By trigonometry,

EXAMPLE 3. Find sin5 θ cos2 θ dθ.

Solution.

EXAMPLE 4. Find cot4 x dx.

Solution.

EXAMPLE 5. Find sin2 θ cos4 θ .

Solution. See §13—4, II, below, for trigonometric substitutions:

13—4. Summary of Trigonometric Reductions.

I. To find sinm x cosn x dx, when either m or n is a positive, odd whole number, we transform the expression to be integrated by means of the relation sin2 θ = 1 — cos2 θ, or cos2 θ = 1 — sin2 θ, in such a way that we may apply the standard form

II. To find sinm x cosn x dx, when both m and n are positive even whole numbers, we make use of the trigonometric relations:

III. To find sin mx cos nx dx, sin mx sin nx dx, or cos mx cos nx dx, we use standard reduction formulas; the proofs of these formulas are not given here, but they are based upon the addition formulas of trigonometry, that is, sin

EXERCISE 13—2

Find the following:

1. cos2 θ dθ

2. cos3 θ dθ

3. sin5 θ

4. cos2 θ sin2 θ dθ

5. sin4 θ dθ

6. cot2 θ

7. sin3 θ cos θ dθ

8. sin2 θ cos θ dθ

9. cos3 θ sin θ

10. tan3 θ dθ

11. sin3 θ cos2 θ

12. (tan 2θ)2

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