Question

Use a computer algebra system to evaluate the definite integral. $$ \int_{0}^{\pi / 4} \sin 2 \theta \sin 3 \theta d \theta $$

Short answer

The value of the definite integral \(\int_{0}^{\pi / 4} \sin 2 \theta \sin 3 \theta d \theta\) is computed to be \((-1/2)[\sin(\pi / 4) + (1/5)\sin(5 \cdot \pi / 4)]\).

Step-by-step solution

Change the form of the integrand

Use the product-to-sum identity, which states that \( \sin a \sin b = (1/2)[\cos(a - b) - \cos(a + b)] \). Substituting \( a = 2\theta \) and \( b = 3\theta \), we can simplify the integrand to (1/2)[\cos(-\theta) - \cos(5\theta)]. So, the task is to compute the integral \(\int_{0}^{\pi / 4} [(1/2)(\cos(-\theta)- \cos(5\theta))] d\theta \).

Split and compute integral

The integral of a difference of two functions is the difference of their integrals. Hence, the task now becomes to compute \((1/2)[\int_{0}^{\pi / 4} \cos(-\theta) d\theta - \int_{0}^{\pi / 4} \cos(5\theta) d\theta]\). The integral of \(\cos(ax)\) is \(\frac{1}{a}\sin(ax)\), using which we find \((1/2)[\sin(-\theta)|_{0}^{\pi / 4} - (1/5)\sin(5\theta)|_{0}^{\pi / 4}]\)

Evaluate the limits

Evaluate above expression over the limits \(0\) to \(\pi / 4\). This gives us \((1/2)[\sin(-\pi / 4) - \sin(0) - (1/5)(\sin(5 \cdot \pi / 4) - \sin(0))]\). Simplifying this expression yields the result.