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.
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Step-by-Step Solution
Step 1: Rewrite the integral
The given integral involves powers of sine and cosine. To simplify it, we can rewrite (cosx)3 as (cosx)2cosx. This allows us to use the identity (cosx)2=1−(sinx)2.
Step 2: Apply trigonometric identity
Substitute (cosx)2=1−(sinx)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=sinx. Then, the differential du=cosxdx. We also need to change the limits of integration according to the substitution. When x=0, u=sin0=0. When x=π/2, u=sin(π/2)=1. The integral becomes a polynomial in u.
Step 4: Expand and integrate
Expand the integrand u4(1−u2) to u4−u6. Then, integrate term by term using the power rule for integration, ∫undu=n+1un+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.