Video: Integrating Trigonometric Functions

Determine ∫ 8 cos (π‘₯/5) dπ‘₯.

01:56

Video Transcript

Determine the integral of eight cos of π‘₯ over five with respect to π‘₯.

The question wants us to calculate the integral of a trigonometric function. And we recall that the derivative with respect to π‘₯ of the sin of π‘₯ is equal to the cos of π‘₯. Another way of saying this is the sin of π‘₯ is an antiderivative of the cos of π‘₯. But we want to calculate an antiderivative of eight multiplied by the cos of π‘₯ over five. So, let’s try to manipulate this equation to try and find an antiderivative of eight multiplied by the cos of π‘₯ over five.

We’ll start by multiplying both sides of our equation by eight. By using our rules of derivatives, we can bring eight inside our derivative. Next, we want to change our angle to be π‘₯ divided by five. We recall, for a constant π‘Ž, if we instead want to take the derivative of the sin of π‘Žπ‘₯, then we get π‘Ž multiplied by the cos of π‘Žπ‘₯. So, we’ve shown that eight multiplied by the sin of π‘Žπ‘₯ is an antiderivative of eight π‘Ž cos of π‘Žπ‘₯, or the integral of eight π‘Ž cos of π‘Žπ‘₯ with respect to π‘₯ is equal to eight sin of π‘Žπ‘₯ plus our constant of integration 𝑐.

The question wants us to calculate the integral of eight cos π‘₯ over five with respect to π‘₯. And we see that the cos of π‘₯ over five is equal to the cos of one-fifth multiplied by π‘₯. So, we’ll set our value of π‘Ž to be equal to one-fifth. So, by setting π‘Ž equal to one-fifth, we’ve shown the integral of eight cos of π‘₯ over five with respect to π‘₯ is equal to eight divided by one-fifth multiplied by the sin of one-fifth π‘₯ plus our constant of integration 𝑐.

Instead of dividing eight by one-fifth, we will multiply eight by the reciprocal of one-fifth. This gives us eight multiplied by five sin of π‘₯ over five plus 𝑐. And we can evaluate eight multiplied by five to be 40. Therefore, we’ve shown the integral eight cos π‘₯ over five with respect to π‘₯ is equal to 40 sin π‘₯ over five plus our constant of integration 𝑐.

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