Evaluate the definite integral $\int_{0}^{\pi/2} \sin^3(x) \cos(x) dx$.

Question

Accepted Answer

We can solve this using the method of substitution.

**Step 1: Choose substitution**
Let $u = \sin(x)$.
Then, the derivative is $du = \cos(x) dx$.

**Step 2: Change the limits of integration**
- When lower limit $x = 0$, $u = \sin(0) = 0$.
- When upper limit $x = \pi/2$, $u = \sin(\pi/2) = 1$.

**Step 3: Substitute and integrate**
$$\int_{0}^{1} u^3 du$$

Using the power rule $\int u^n du = \frac{u^{n+1}}{n+1}$:
$$\left[ \frac{u^4}{4} \right]_{0}^{1} = \frac{1^4}{4} - \frac{0^4}{4} = \frac{1}{4}$$

Thus, the value of the integral is $\frac{1}{4}$.

Evaluate the definite integral $\int_{0}^{\pi/2} \sin^3(x) \cos(x) dx$. - DDCET Q&A