### Video Transcript

Determine the indefinite integral of five π₯ minus 12 multiplied by sin π₯ with respect to π₯.

The first thing we should notice is that weβve been asked to integrate a product of functions. We have the integral of five π₯ minus 12 multiplied by sin π₯. And this suggests that weβre going to need to use the method of integration by parts. The integration by parts formula tells us that the integral of π’ multiplied by dπ£ by dπ₯ with respect π₯ is equal to π’π£ minus the integral of π£ multiplied by dπ’ by dπ₯ with respect to π₯. The integration by parts formula is essentially to integration what the product rule is to differentiation.

What we need to do then is to decide which of the functions in our product should be π’ and which should be dπ£ by dπ₯. Now, from the integration by parts formula, we can see that later on, weβre going to need to integrate π£ multiplied by dπ’ by dπ₯. So weβd like to choose π’ to be something that becomes simpler when we find its derivative. Of the two functions five π₯ minus 12 and sin π₯, five π₯ minus 12 does indeed become simpler when we differentiate. It gives a polynomial of lower degree. In fact, it just gives a constant. Whereas sin π₯ gives another trigonometric function. So five π₯ minus 12 becomes simpler and sin π₯ doesnβt.

This suggests, then, that we want to let π’ equal the function five π₯ minus 12. In fact, if weβre struggling to decide, there is a helpful acronym we can use to assign our function π’. Itβs the word LIATE. The letter L stands for logarithmic. So the first type of function we should assign to be π’ is a logarithmic function. I stands for inverse trigonometric. A stands for algebraic. T stands for trigonometric. And, finally, E stands for exponential. We can see that algebraic functions, in this case five π₯ minus 12, do indeed appear before trigonometric functions, in this case sin π₯, in the order given by LIATE. So we were correct in assigning π’ to be the function five π₯ minus 12. This means then that our function dπ£ by dπ₯ must be equal to sin π₯.

We need to find dπ’ by dπ₯, which weβve already seen to be five. And then, we need to find π£ by integrating sin π₯. Now, the integral of sin π₯ with respect to π₯ is negative cos π₯ plus a constant of integration, if performing an indefinite integral. But we wonβt include it at this stage. Weβll just make sure that we do add a constant of integration when weβre finished. Now, weβre ready to substitute into our integration by parts formula.

We have that the integral of π’ times dπ£ by dπ₯ is equal to π’ times π£. Thatβs five π₯ minus 12 times negative cos π₯ minus the integral of π£ times dπ’ by dπ₯, or dπ’ by dπ₯ times π£, which are written as five times negative cos π₯. In our first term, π’π£, we can distribute the negative sin with cos π₯ into the first set of parentheses, giving negative five π₯ plus 12 multiplied by cos π₯. And then, for our integral, we can bring out a factor of negative five making it plus five times the integral of cos π₯ with respect to π₯. The integral of cos π₯ is sin π₯. So we have plus five sin π₯. And we include our constant of integration, which weβll call π.

By applying integration by parts then, weβve found that the indefinite integral of five π₯ minus 12 multiplied by sin π₯ with respect to π₯ is equal to negative five π₯ plus 12 multiplied by cos π₯ plus five sin π₯ plus π.