Question

Use a computer algebra system to evaluate the definite integral. $$ \int_{0}^{\pi / 2} \sin ^{6} x d x $$

Short answer

The definite integral of \(\sin ^{6} x\) from 0 to \(\pi / 2\) is approximately 0.02083.

Step-by-step solution

Apply the reduction formula

The reduction formula, \(\sin ^{2 n} x =\frac{1}{2^n. n!} . \sum _{k=0} ^n \binom {n}{k} (2n-2k)! (-4)^{n-k}\), should be applied first to break down \(\sin ^6 x\) into a simpler form.

Use the algebra system to integrate

After breaking down the function using the reduction formula, the integral \(\int _0 ^{\pi /2} \sin ^6 x dx\) can be evaluated. The antiderivative at the limits of integration, from 0 to \(\pi/2\), must then be calculated.

Calculate the definite integral

Finally, subtract the antiderivative at the lower limit (0) from the antiderivative at the upper limit \(\pi/2\) . The result will be the value for the definite integral.