Question

Find \(\int_{0}^{\frac{\pi}{2}} cos^3(x).cos(2x)dx\).

(a) 0

(b) 5

(c) 87

(d) ^-16⁄105

This question was addressed to me in a national level competition.

This intriguing question originated from Integral Reduction Formula topic in chapter Integral Calculus of Engineering Mathematics

Answer 1

Correct choice is (d) ^-16⁄105

Best explanation: Rewriting the function as

\(\int_{0}^{\frac{\pi}{2}} cos^3(x).cos(2x)dx=2\int_{0}^{\frac{\pi}{2}} cos^7(x)dx-\int_{0}^{\frac{\pi}{2}} cos^5(x)dx\)

We now use the formula

\(\int_0^{\frac{\pi}{2}} cos^n(x)dx=\frac{(n-1).(n-3)…2}{n.(n-2)…\times 1}\)

\(=2.\frac{6.4.2}{7.5.3.1}-\frac{4.2}{5.3.1}=\frac{4.2.-2}{5.3.1.7}\)

=\( \frac{16}{105}\)