Evaluate the definite integral: integral from 0 to pi/2 of sin^4(x)*cos^3(x) dx.

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Step-by-Step Solution

Step 1: Rewrite the integrand

To evaluate this integral, we first rewrite the integrand. We separate one factor of \cos(x) and use the identity \cos^2(x) = 1 - \sin^2(x) to express the remaining \cos^2(x) in terms of \sin(x)$. This prepares the integral for a substitution.

Step 2: Apply substitution

Now, we apply a substitution. Let u = \sin(x)$. Then, the differential du becomes \cos(x) \, dx$. We also need to change the limits of integration according to the substitution. When x = 0$, u = \sin(0) = 0$. When x = \frac{\pi}{2}$, u = \sin\left(\frac{\pi}{2}\right) = 1$.

Step 3: Substitute into the integral

Substitute u = \sin(x) and du = \cos(x) \, dx into the integral, along with the new limits of integration. The integral now becomes a polynomial in u$.

Step 4: Expand the integrand

Expand the integrand by multiplying u^4 by (1 - u^2)$. This simplifies the expression, making it easier to integrate term by term.

Step 5: Integrate term by term

Now, integrate each term of the polynomial with respect to u$. We use the power rule for integration, which states that \int u^n \, du = \frac{u^{n+1}}{n+1} + C$.

Step 6: Evaluate the definite integral

Finally, we evaluate the definite integral by substituting the upper limit (u=1) and the lower limit (u=0) into the antiderivative and subtracting the results. The terms with 0 evaluate to 0$, leaving us with a simple subtraction of fractions.