Evaluate the integral.
-$$\int \cos \frac { \theta } { 2 } \cos \frac { \theta } { 5 } d \theta$$
A) $\frac { 5 } { 3 } \sin \frac { 3 } { 10 } \theta + \frac { 5 } { 7 } \sin \frac { 7 } { 10 } \theta + C$
B) $\frac { 5 } { 3 } \sin \frac { 3 } { 10 } \theta - \frac { 5 } { 7 } \sin \frac { 7 } { 10 } \theta + C$
C) $\frac { 5 } { 3 } \cos \frac { 3 } { 10 } \theta + \frac { 5 } { 7 } \cos \frac { 7 } { 10 } \theta + C$
D) $\frac { 5 } { 3 } \sin 3 \theta + \frac { 5 } { 7 } \sin 7 \theta + C$

Question

Quizplus · Accepted Answer

The integral of $\cos \frac{\theta}{2} \cos \frac{\theta}{5}$ can be solved using the product-to-sum formulas for trigonometric functions. The formula $\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]$ simplifies the integral into a form that can be directly integrated. Applying this formula, the integral becomes a sum of two simpler integrals, each of which leads to a sine function with coefficients that match the options in choice A.

Evaluate the Integral ∫cos⁡θ2cos⁡θ5dθ\int \cos \frac { \theta } { 2 } \cos \frac { \theta } { 5 } d \theta∫cos2θ​cos5θ​dθ

Evaluate the Integral $\int \cos \frac { \theta } { 2 } \cos \frac { \theta } { 5 } d \theta$