Question Video: Finding the Integration of a Function Involving Trigonometric Functions Using Integration by Substitution Mathematics

Video Transcript

Determine the integral of negative 24𝑥 cubed plus 30 sin six 𝑥 times negative six 𝑥 to the fourth power minus five cos of six 𝑥 to the fifth power with respect to 𝑥.

To evaluate this integral, we need to spot that negative 24𝑥 cubed plus 30 sin six 𝑥 is the derivative of the inner part of this composite function negative six 𝑥 to the fourth power minus five cos six 𝑥. This tells us we can use integration by substitution to evaluate this integral. We’ll let 𝑢 be the inner function in our composite function, and then we use the general result for the derivative of cos 𝑎𝑥.

And we see that d𝑢 by d𝑥, the derivative of 𝑢 with respect to 𝑥, is negative 24𝑥 cubed plus 30 sin six 𝑥. Remember, d𝑢 by d𝑥 is not a fraction, but we do treat it a little like one when performing integration by substitution. And we see that this is equivalent to saying that d𝑢 is equal to negative 24𝑥 cubed plus 30 sin six 𝑥 d𝑥. We, therefore, replace negative 24𝑥 cubed plus 30 sin six 𝑥d𝑥 with d𝑢. And we replace negative six 𝑥 to the fourth power minus five cos six 𝑥 with 𝑢.

And we see that our integral becomes really nice. It’s the integral of 𝑢 to the fifth power d𝑢. Well, the antiderivative of 𝑢 to the fifth power is 𝑢 to the sixth power over six. So, the integral of 𝑢 to the fifth power d𝑢 is 𝑢 to the sixth power over six plus the constant of integration 𝑐. Remember though, our integral’s in terms of 𝑥, so we replace 𝑢 with negative six 𝑥 to the fourth power minus five cos of six 𝑥.

And we’ve evaluated our integral. It’s a sixth of negative six 𝑥 to the fourth power minus five cos of six 𝑥 to the sixth power plus 𝑐.