# Question Video: Finding the Integration of a Function Involving a Trigonometric Function Mathematics • Higher Education

Determine β«(4π₯ β 5 cos 6π₯) dπ₯.

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### Video Transcript

Determine the integral of four π₯ minus five times the cos of six π₯ with respect to π₯.

The question wants us to determine the integral of the difference between two functions. Our integrand is the difference between four π₯ and five times the cos of six π₯. And we know how to integrate each of these terms in our integrand separately. So, to determine this integral, weβll just split our integral into two integrals which we can evaluate.

First, we recall that the integral of a difference between two functions is equal to the difference between their integrals. Using this, we can rewrite our integral as the integral of four π₯ with respect to π₯ minus the integral of five times the cos of six π₯ with respect to π₯. And we could now evaluate each of these integrals separately.

First, to integrate four π₯ with respect to π₯, weβll use the power rule for integration. And we recall this tells us, for constants π and π where π is not equal to negative one, to integrate ππ₯ to the πth power with respect to π₯, we add one to our exponent and divide by this new exponent. Then, we add our constant of integration π. So, by thinking of four π₯ as four π₯ to the first power, we can evaluate our first integral as four π₯ squared divided by two plus the constant of integration weβll call π one.

We then need to evaluate our second integral. We can see that this is a trigonometric integral. And this is a standard trigonometric integral which we should commit to memory. For constants π and π where π is not equal to zero, the integral of π times the cos of ππ₯ with respect to π₯ is equal to π times the sin of ππ₯ divided by π plus the constant of integration π. So, by using our standard trigonometric integral rule, we get the integral of five times the cos of six π₯ with respect to π₯ is equal to five times the sin of six π₯ divided by six plus the constant of integration π two.

And remember, since we were subtracting this integral, we need to subtract the result. Canceling the shared factor of two in the numerator and the denominator of four π₯ squared, distributing the negative over our parentheses, and rearranging, we get two π₯ squared minus five over six sin six π₯ plus π one minus π two.

And we can do one more thing to simplify this answer. Since π one and negative π two are both constants, we can combine both of these into a new constant, which we will call π. Therefore, we have shown the integral of four π₯ minus five times the cos of six π₯ with respect to π₯ is equal to two π₯ squared minus five over six times the sin of six π₯ plus π.