Integrals Involving sin(x) and cos(x) with odd power

Integrals Involving sin(x) and cos(x) with Odd Power

Tutorial to find integrals involving the product of powers of sin(x) and cos(x) with one of the two having an odd power. Exercises with answers are at the bottom of the page.

In what follows, C is the constant of integration.

Example 1: Evaluate the integral

sin3(x) cos2(x) dx


Solution to Example 1:

The main idea is to rewrite the integral writing the term with the odd power as the product of a term with power 1 and a term with an even power. Example: sin3(x) = sin2(x) sin(x). Hence the given integral may be written as follows:

sin3(x) cos2(x) dx = sin2(x) cos2(x) sin(x) dx

We now use the identity sin2(x) = 1 - cos2(x) and rewrite the given integral as follows:

sin3(x) cos2(x) dx = (1 - cos2(x)) cos2(x) sin(x) dx

We now let u = cos(x), hence du/dx = -sin(x) or -du = sin(x)dx and substitute in the given intergral to obtain

sin3(x) cos2(x) dx = -(1 - u2) u2 du

Expand and calculate the integral on the right

sin3(x) cos2(x) dx = u4 - u2 du

= (1/5)u
5 - (1/3)u3 + C

Substitute u by cos(x) to obtain

sin3(x) cos2(x) dx = (1/5)cos5(x) - (1/3)cos3(x) + C

Example 2: Evaluate the integral

sin12(x) cos5(x) dx


Solution to Example 2:

Rewrite cos5(x) as follows cos5(x) = cos4(x) cos(x). Hence the given integral may be written as follows:

sin12(x) cos5(x) dx = sin12(x) cos4(x) cos(x) dx

We now use the identity cos2(x) = 1 - sin2(x) to rewrite cos4(x) in terms of power of sin(x) and rewrite the given integral as follows:

sin12(x) cos5(x) dx = sin12(x) (1 - sin2(x))2 cos(x) dx

We now let u = sin(x), hence du/dx = cos(x) or du = cos(x)dx and substitute in the given intergral to obtain

sin12(x) cos5(x) dx = u12 (1 - u2)2 du

Expand and calculate the integral on the right

sin12(x) cos5(x) dx = u12 (1 + u4 - 2u2) du

=
(u16 - 2u14 + u12 ) du

= (1/17)u
17 - (2/15)u15 + (1/13)u13 + C

Substitute u by sin(x) to obtain

sin3(x) cos2(x) dx = (1/17)sin17(x) - (2/15)sin15(x) + (1/13)sin13(x) + C

Exercises: Evaluate the following integrals.

1. cos3(x) sin2(x) dx

2. sin3(x) cos14(x) dx



Answers to Above Exercises

1. - (1/5)sin5(x) + (1/3)sin3(x)

2. (1/17)cos17(x) - (1/15)cos15(x)

More references on integrals and their applications in calculus.

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