Evaluate ∫₀^(π/2) (sin x)^4 (cos x)^3 dx using reduction formulas.
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Step-by-Step Solution
Step 1: Rewrite the integral
The integral involves powers of sinx and cosx. Since the power of cosx is odd, we can separate one cosx term and convert the remaining even power of cosx into sinx using the identity cos2x=1−sin2x.
Step 2: Substitute using trigonometric identity
Substitute cos2x=1−sin2x into the integral. This transforms the integral into a form suitable for a substitution.
Step 3: Perform substitution
Let u=sinx. Then, du=cosxdx. We also need to change the limits of integration. When x=0, u=sin0=0. When x=π/2, u=sin(π/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, ∫undu=n+1un+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.