VARIATION: Evaluate ∫₀^(π/2) (sin x)^4 (cos x)^3 dx using reduction formulas. Find the numerical value if exact solution is not possible.

Step 1: Rewrite the integral

The given integral involves powers of sine and cosine. To simplify it, we can rewrite $(\cos x)^3$ as $(\cos x)^2 \cos x$. This allows us to use the identity $(\cos x)^2 = 1 - (\sin x)^2$.

Step 2: Apply trigonometric identity

Substitute $(\cos x)^2 = 1 - (\sin x)^2$ into the integral. This transforms the integral into a form that can be solved using a simple substitution.

Step 3: Perform substitution

Let $u = \sin x$. Then, the differential $du = \cos x dx$. We also need to change the limits of integration according to the substitution. When $x=0$, $u=\sin 0 = 0$. When $x=\pi/2$, $u=\sin (\pi/2) = 1$. The integral becomes a polynomial in $u$.

Step 4: Expand and integrate

Expand the integrand $u^4 (1 - u^2)$ to $u^4 - u^6$. Then, integrate term by term using the power rule for integration, $\int u^n du = \frac{u^{n+1}}{n+1}$.

Step 5: Evaluate the definite integral

Substitute the upper limit $u=1$ and the lower limit $u=0$ into the integrated expression. Subtract the value at the lower limit from the value at the upper limit.

Step 6: Calculate the final value

Perform the subtraction of the fractions to get the final numerical value of the integral.

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