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

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.

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