# Video: Integrating Trigonometric Functions

Determine ∫ (−8 sin 8𝑥 − 7 cos 5𝑥) d𝑥.

01:55

### Video Transcript

Determine the indefinite integral of negative eight sin of eight 𝑥 minus seven cos of five 𝑥 evaluated with respect to 𝑥.

In this question, we’re looking to integrate the sum of two functions in 𝑥. We begin by recalling the fact that the integral of the sum of two or more functions is actually equal to the sum of the integrals of those respective functions. So we can write our problem as the integral of negative eight sin of eight 𝑥 with respect to 𝑥 plus the integral of negative seven cos of five 𝑥 d𝑥. We also know that we can take any constant factors outside of the integral and focus on integrating each expression in 𝑥. So we can rewrite our problem further as negative eight times the integral of sin of eight 𝑥 with respect to 𝑥 minus seven times the integral of cos of five 𝑥 with respect to 𝑥.

Next, we recall the general results for the integral of sine and cosine. The integral of sin of 𝑎𝑥 is negative one over 𝑎 cos of 𝑎𝑥 plus 𝑐. And the indefinite integral of cos of 𝑎𝑥 is one over 𝑎 sin of 𝑎𝑥 plus 𝑐. We integrate each function, respectively, and we see that the integral of sin of eight 𝑥 is negative one-eighths cos of eight 𝑥 plus 𝐴. And the indefinite integral of cos of five 𝑥 is a fifth sin of five 𝑥 plus 𝐵. And I’ve chosen 𝐴 and 𝐵, as opposed to just one value of 𝑐, to show that these are actually different constants.

Our final step is to distribute the parentheses. Negative eight times negative one-eighths cos of eight 𝑥 is just cos of eight 𝑥. Negative seven times one-fifth of sin of five 𝑥 is a negative seven-fifths sin of five 𝑥. Finally, we multiply negative eight by 𝐴 and negative seven by 𝐵. And we end up with this new constant 𝐶. And we found that the integral we required is cos of eight 𝑥 minus seven-fifths of sin of five 𝑥 plus 𝐶.