TRIGONOMETRIC INTEGRALS - Special Methods of Integration - The Calculus Primer

The Calculus Primer (2011)

Part XIII. Special Methods of Integration


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 image sin2 x dx.

Solution. By trigonometry, sin2 image


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

since sin 2x = 2 sin x cos x.

EXAMPLE 2. Find image sin3 θ dθ.

Solution. By trigonometry,


EXAMPLE 3. Find image sin5 θ cos2 θ dθ.


EXAMPLE 4. Find image cot4 x dx.


EXAMPLE 5. Find image sin2 θ cos4 θ .

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

13—4. Summary of Trigonometric Reductions.

I. To find image 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 image 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 image sin mx cos nx dx, image sin mx sin nx dx, or image 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 image


Find the following:

1. image cos2 θ dθ

2. image cos3 θ dθ

3. image sin5 θ

4. image cos2 θ sin2 θ dθ

5. imagesin4 θ dθ

6. image cot2 θ

7. image sin3 θ cos θ dθ

8. imagesin2 θ cos θ dθ

9. image cos3 θ sin θ

10. image tan3 θ dθ

11. image sin3 θ cos2 θ

12. image (tan 2θ)2