Antiderivative of f(x) = cos³ x

This problem asks to find the antiderivative F(x) of the function f(x) = cos³ x satisfying the condition F(π) = 0.

Problem

$f (x) = {cos}^{3} x, F (π) = 0$

Solution

Step 1

Represent cos³ x as the product cos² x · cos x. This will isolate the factor cos x, which is the derivative of sin x.

{cos}^{3} x = {cos}^{2} x \cdot cos x

Step 2

Note that d(sin x) = cos x dx. This allows us to bring cos x under the differential sign.

d (sin x) = cos x d x

Step 3

Rewrite the original integral by isolating the factor cos² x and replacing cos x dx with d(sin x):

\int {cos}^{3} x d x = \int {cos}^{2} x d (sin x)

Step 4

Apply the basic trigonometric identity:

{cos}^{2} x = 1 - {sin}^{2} x

Step 5

Substitute this expression into the integral:

\int (1 - {sin}^{2} x) d (sin x)

Step 6

The integral splits into two tabular ones: ∫ dt = t and ∫ t² dt = t³/3. Returning to the variable x, we get:

sin x - \frac{{sin}^{3} x}{3} + C

Step 7

Find the constant C from the condition F(π) = 0. Substitute x = π: sin π = 0, therefore:

sin π - \frac{{sin}^{3} π}{3} + C = 0

Step 8

From this, C = 0. The required antiderivative has the form:

Answer

$F (x) = sin x - \frac{{sin}^{3} x}{3}$

Frequently asked questions

How do I find the antiderivative of f(x) = cos³ x?

Represent cos³ x = cos² x · cos x, bring cos x under the differential sign (since d(sin x) = cos x dx), and apply the basic trigonometric identity cos² x = 1 − sin² x. Then the integral reduces to tabular ones.

Why is the basic trigonometric identity needed?

It allows us to express cos² x in terms of sin x: cos² x = 1 − sin² x. This is necessary so that after the substitution t = sin x, the integral contains only the variable t and becomes tabular.

Why is C = 0 in this problem?

From the condition F(π) = 0. At x = π: sin π = 0 and sin³ π = 0, so both terms vanish. Hence C = 0.

What is special about integrating odd powers of cosine?

For odd powers of cos x (or sin x), use the following technique: separate one factor cos x (or sin x) and bring it under the differential, then express the remaining even power in terms of the opposite function using the basic trigonometric identity.