Video Transcript
Determine the integral of negative 24π₯ cubed plus 30 sin six π₯ times negative six π₯ to the fourth power minus five cos of six π₯ to the fifth power with respect to π₯.
To evaluate this integral, we need to spot that negative 24π₯ cubed plus 30 sin six π₯ is the derivative of the inner part of this composite function negative six π₯ to the fourth power minus five cos six π₯. This tells us we can use integration by substitution to evaluate this integral. Weβll let π’ be the inner function in our composite function, and then we use the general result for the derivative of cos ππ₯.
And we see that dπ’ by dπ₯, the derivative of π’ with respect to π₯, is negative 24π₯ cubed plus 30 sin six π₯. Remember, dπ’ by dπ₯ 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 dπ’ is equal to negative 24π₯ cubed plus 30 sin six π₯ dπ₯. We, therefore, replace negative 24π₯ cubed plus 30 sin six π₯dπ₯ with dπ’. And we replace negative six π₯ to the fourth power minus five cos six π₯ with π’.
And we see that our integral becomes really nice. Itβs the integral of π’ to the fifth power dπ’. Well, the antiderivative of π’ to the fifth power is π’ to the sixth power over six. So, the integral of π’ to the fifth power dπ’ is π’ to the sixth power over six plus the constant of integration π. Remember though, our integralβs in terms of π₯, so we replace π’ with negative six π₯ to the fourth power minus five cos of six π₯.
And weβve evaluated our integral. Itβs a sixth of negative six π₯ to the fourth power minus five cos of six π₯ to the sixth power plus π.
