Evaluate the Integral

Problem

(∫_0^π/2)(cos(x)*d(x))

Solution

1. Use a trigonometric identity to rewrite the integrand by separating one cos(x) factor.

(∫_0^π/2)(cos(x)*cos(x)*d(x))

2. Express the even power of cosine in terms of sine using the identity cos(x)=1−sin(x)

(∫_0^π/2)((1−sin(x))2*cos(x)*d(x))

3. Apply u-substitution by letting u=sin(x) which implies d(u)=cos(x)*d(x)

u=sin(x)

4. Change the limits of integration based on the substitution.

When *x=0,u=sin(0)=0

When *x=π/2,u=sin(π/2)=1

5. Substitute and expand the polynomial expression within the integral.

(∫_0^1)((1−u2)2*d(u))

(∫_0^1)((1−2*u2+u4)*d(u))

6. Integrate the polynomial term by term using the power rule.

[u−(2*u3)/3+(u5)/5]10

7. Evaluate the definite integral at the upper and lower bounds.

(1−2/3+1/5)−(0−0+0)

15/15−10/15+3/15

8/15

Final Answer

(∫_0^π/2)(cos(x)*d(x))=8/15

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