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Question Video: Integrating Trigonometric Functions Mathematics • Second Year of Secondary School

Determine ∫(−8 sin 8𝑥 − 7 cos 5𝑥) d𝑥.

02:20

Video Transcript

Determine the indefinite integral with respect to 𝑥 of negative eight times the sin of eight 𝑥 minus seven times the cos of five 𝑥.

In this question, we’re looking to integrate the sum of two functions of 𝑥. We begin by recalling that the integral of the sum of two or more functions is equal to the sum of the integrals of those respective functions. And so we can write our integral as the integral of negative eight sin of eight 𝑥 with respect to 𝑥 plus the integral of negative seven cos of five 𝑥 d𝑥. We also know that we can take any constant factors outside of the integrals and focus on integrating each expression in 𝑥. And this means we can rewrite our integrals as negative eight times the integral with respect to 𝑥 of sin eight 𝑥 minus seven times the integral of cos five 𝑥 d𝑥.

Next, we recall the general results for the integrals of sin 𝑎𝑥 and cos 𝑎𝑥. The indefinite integral with respect to 𝑥 of sin of 𝑎𝑥 is equal to negative one over 𝑎 times cos 𝑎𝑥 plus the constant of integration 𝑐. And the indefinite integral of cos 𝑎𝑥 with respect to 𝑥 is one over 𝑎 sin 𝑎𝑥 plus the constant 𝑐. In our case, the constant 𝑎 is eight in the first integral and five in the second integral. And applying these results to our integrals, we see that the integral of sin of eight 𝑥 is negative one over eight cos eight 𝑥 plus the constant 𝐴. And the integral with respect to 𝑥 of cos of five 𝑥 is one over five times the sin of five 𝑥 plus 𝐵. Note that we’ve chosen 𝐴 and 𝐵 as the constant of integration, not just a single value of 𝑐, to show that these are actually different constants.

Our final step is to distribute the parentheses. Negative eight times negative one over eight. cos of eight 𝑥 is just the cos of eight 𝑥. And negative seven times one over five sin five 𝑥 is negative seven over five sin five 𝑥. And finally, we multiply negative eight by 𝐴 and negative seven by 𝐵. And since we don’t know the values of 𝐴 and 𝐵, we choose to represent this as a single constant 𝑐. We found then that the integral we required is the cos of eight 𝑥 minus seven over five times the sin of five 𝑥 plus the constant 𝑐.

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