Video: Integrating Trigonometric Functions

Video Transcript

Determine the integral of negative seven times the sin of seven 𝑥 with respect to 𝑥.

The question wants us to determine the integral of a trigonometric function. And this is actually in the form of a standard trigonometric integral which we should commit to memory. For constants 𝑎 and 𝑛, where 𝑎 is not equal to zero, the integral of 𝑛 times the sin of 𝑎𝑥 with respect to 𝑥 is equal to negative 𝑛 times the cos of 𝑎𝑥 divided by 𝑎 plus a constant of integration 𝑐.

In our case, we can see we’re multiplying our sine function by negative seven. So we’ll set 𝑛 equal to negative seven. And we can see we’re taking the sin of seven 𝑥, so we’ll set our value of 𝑎 equal to seven. So we can evaluate this integral by just substituting the values of 𝑛 is equal to negative seven and 𝑎 is equal to seven into our integral rule. We get negative one times negative seven times the cos of seven 𝑥 divided by seven plus a constant of integration 𝑐.

And we can simplify this answer. First, negative one multiplied by negative seven is equal to seven. Next, we can cancel the shared factor of seven in our numerator and our denominator. And this gives us our final answer, the cos of seven 𝑥 plus 𝑐. Therefore, we’ve shown the integral of negative seven times the sin of seven 𝑥 with respect to 𝑥 is equal to the cos of seven 𝑥 plus our constant of integration 𝑐.