Antiderivative of f(x) = sin³ x

This problem asks to find the antiderivative F(x) of the function f(x) = sin³ x satisfying the condition F(π/2) = 0.

Problem

$f (x) = {sin}^{3} x, F (π / 2) = 0$

Solution

Step 1

Represent sin³ x as the product sin² x · sin x. This will isolate the factor sin x, which is the derivative of cos x (up to sign).

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

Step 2

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

d (cos x) = - sin x d x

Step 3

Rewrite the original integral by isolating the factor sin² x and replacing sin x dx with −d(cos x):

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

Step 4

Apply the basic trigonometric identity:

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

Step 5

Substitute this expression into the integral:

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

Step 6

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

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

Step 7

Find the constant C from the condition F(π/2) = 0. Substitute x = π/2: cos(π/2) = 0, therefore:

- cos \frac{π}{2} + \frac{{cos}^{3} \frac{π}{2}}{3} + C = 0

Step 8

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

Answer

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

Frequently asked questions

How do I find the antiderivative of f(x) = sin³ x?

Represent sin³ x = sin² x · sin x, bring sin x under the differential sign (since d(cos x) = −sin x dx, i.e. sin x dx = −d(cos x)), and apply the basic trigonometric identity sin² x = 1 − cos² x. Then the integral reduces to tabular ones in cos x.

Why is the basic trigonometric identity needed?

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

Why is C = 0 in this problem?

From the condition F(π/2) = 0. At x = π/2: cos(π/2) = 0 and cos³(π/2) = 0, so both terms vanish. Hence C = 0.

What is special about integrating odd powers of sine?

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