Evaluate the definite integral: integral from 0 to pi/2 of sin4(x)∗cos3(x)dx.
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Step-by-Step Solution
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 cos2(x)=1 - sin2(x) to express the remaining cos2(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=2π ,u=sin(2π)=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$.
Step 4: Expand the integrand
Expand the integrand by multiplying u4 by (1 - u2)$. 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 ∫undu=n+1un+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.