Solved Exercise: Integral Example 3

Solve the antiderivative:

$$ \int \sin^3 (x) \cdot \cos(x) \ dx $$

To solve this indefinite integral, use the substitution method.

Let t = \sin(x) as the substitution variable:

$$ t = \sin(x) $$

Differentiate both sides with respect to \( x \):

$$ \frac{dt}{dx} = \cos(x) $$

which gives:

$$ dt = \cos(x) \, dx $$

Substitute t = \sin(x) into the integral.

Since \( t = \sin(x) \), it follows that \( t^3 = \sin^3(x) \).

Thus, the integral becomes:

$$ \int t^3 \cdot \cos(x) \, dx $$

Substitute \( dt = \cos(x) \, dx \):

$$ \int t^3 \, dt $$

This transforms the original integral into an antiderivative of a power.

The indefinite integral of \( t^n \) is given by:

$$ \int t^n \, dt = \frac{t^{n+1}}{n+1} + c $$

Applying this formula:

$$ \int t^3 \, dt = \frac{t^{3+1}}{3+1} + c $$

$$ = \frac{t^4}{4} + c $$

Finally, substitute back \( t = \sin(x) \):

$$ \int t^3 \, dt = \frac{\sin^4(x)}{4} + c $$

This is the solution of the indefinite integral.