Evaluate ∫₀^(π/2) (sin x)^4 (cos x)^3 dx using reduction formulas.

Step 1: Rewrite the integral

The integral involves powers of $\sin x$ and $\cos x$. Since the power of $\cos x$ is odd, we can separate one $\cos x$ term and convert the remaining even power of $\cos x$ into $\sin x$ using the identity $\cos^2 x = 1 - \sin^2 x$.

Step 2: Substitute using trigonometric identity

Substitute $\cos^2 x = 1 - \sin^2 x$ into the integral. This transforms the integral into a form suitable for a substitution.

Step 3: Perform substitution

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

Step 4: Expand and integrate

Expand the integrand and 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 and lower limits of integration into the integrated expression and subtract the lower limit value from the upper limit value to find the definite integral's result.

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