Determine the indefinite integral of five 𝑥 minus 12 multiplied by sin 𝑥 with respect to 𝑥.
The first thing we should notice is that we’ve been asked to integrate a product of functions. We have the integral of five 𝑥 minus 12 multiplied by sin 𝑥. And this suggests that we’re going to need to use the method of integration by parts. The integration by parts formula tells us that the integral of 𝑢 multiplied by d𝑣 by d𝑥 with respect 𝑥 is equal to 𝑢𝑣 minus the integral of 𝑣 multiplied by d𝑢 by d𝑥 with respect to 𝑥. 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 𝑢 and which should be d𝑣 by d𝑥. Now, from the integration by parts formula, we can see that later on, we’re going to need to integrate 𝑣 multiplied by d𝑢 by d𝑥. So we’d like to choose 𝑢 to be something that becomes simpler when we find its derivative. Of the two functions five 𝑥 minus 12 and sin 𝑥, five 𝑥 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 𝑥 gives another trigonometric function. So five 𝑥 minus 12 becomes simpler and sin 𝑥 doesn’t.
This suggests, then, that we want to let 𝑢 equal the function five 𝑥 minus 12. In fact, if we’re struggling to decide, there is a helpful acronym we can use to assign our function 𝑢. It’s the word LIATE. The letter L stands for logarithmic. So the first type of function we should assign to be 𝑢 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 𝑥 minus 12, do indeed appear before trigonometric functions, in this case sin 𝑥, in the order given by LIATE. So we were correct in assigning 𝑢 to be the function five 𝑥 minus 12. This means then that our function d𝑣 by d𝑥 must be equal to sin 𝑥.
We need to find d𝑢 by d𝑥, which we’ve already seen to be five. And then, we need to find 𝑣 by integrating sin 𝑥. Now, the integral of sin 𝑥 with respect to 𝑥 is negative cos 𝑥 plus a constant of integration, if performing an indefinite integral. But we won’t include it at this stage. We’ll just make sure that we do add a constant of integration when we’re finished. Now, we’re ready to substitute into our integration by parts formula.
We have that the integral of 𝑢 times d𝑣 by d𝑥 is equal to 𝑢 times 𝑣. That’s five 𝑥 minus 12 times negative cos 𝑥 minus the integral of 𝑣 times d𝑢 by d𝑥, or d𝑢 by d𝑥 times 𝑣, which are written as five times negative cos 𝑥. In our first term, 𝑢𝑣, we can distribute the negative sin with cos 𝑥 into the first set of parentheses, giving negative five 𝑥 plus 12 multiplied by cos 𝑥. And then, for our integral, we can bring out a factor of negative five making it plus five times the integral of cos 𝑥 with respect to 𝑥. The integral of cos 𝑥 is sin 𝑥. So we have plus five sin 𝑥. And we include our constant of integration, which we’ll call 𝑐.
By applying integration by parts then, we’ve found that the indefinite integral of five 𝑥 minus 12 multiplied by sin 𝑥 with respect to 𝑥 is equal to negative five 𝑥 plus 12 multiplied by cos 𝑥 plus five sin 𝑥 plus 𝑐.