Video: Integrating Trigonometric Functions

Determine โˆซ(3 sin 7๐‘ฅ + 4 cos 7๐‘ฅ) d๐‘ฅ.

03:44

Video Transcript

Determine the integral of three sin seven ๐‘ฅ plus four cos seven ๐‘ฅ d๐‘ฅ.

So the first thing you can do is split our expression. And thatโ€™s because if weโ€™re gonna integrate it, we can integrate each term separately. So weโ€™ve got the integral of three sin seven ๐‘ฅ d๐‘ฅ plus the integral of four cos seven ๐‘ฅ d๐‘ฅ. So then, to make our integration simpler, we can take our constant terms. So this leaves us with three multiplied by the integral of sin seven ๐‘ฅ d๐‘ฅ plus four multiplied by the integral of cos seven ๐‘ฅ d๐‘ฅ. And we can do that because they wonโ€™t affect the integration itself. So what Iโ€™m gonna do is deal with the left-hand side first. And what Iโ€™m gonna do is integrate sin seven ๐‘ฅ. And when I integrate sin seven ๐‘ฅ, I get negative one over seven cos seven ๐‘ฅ. And now, Iโ€™d have plus ๐‘. And thatโ€™s the constant of integration. But Iโ€™m gonna add that in at the end.

Now, the reason I got that is because we know that the integral of sin ๐‘Ž๐‘ฅ d๐‘ฅ is equal to negative one over ๐‘Ž cos ๐‘Ž๐‘ฅ plus ๐‘. In our case, the ๐‘Ž was seven. Well, the question is, how do we get that general result? And we can show that using substitution. So by substitution, what we can do is we can say that ๐‘ข is equal to seven ๐‘ฅ. So then, what we can do is differentiate to find d๐‘ข d๐‘ฅ. And when we do that, we get d๐‘ข d๐‘ฅ is equal to seven. And thatโ€™s because if we differentiate seven ๐‘ฅ, we just get seven if we differentiate with respect to ๐‘ฅ. And then, we can say that d๐‘ฅ is equal to one over seven d๐‘ข. And we get that by dealing with d๐‘ข d๐‘ฅ similar to our fraction. However, it definitely is not a fraction. It can just be used in that way in this process.

So then, what weโ€™re gonna get is one-seventh, cause Iโ€™ve taken the constant outside the integration, multiplied by the integral of sin ๐‘ข d๐‘ข. And we know the integral of sin ๐‘ข, because the integral of sin ๐‘ฅ is one of our standard integrals. And the integral of sin ๐‘ฅ is equal to negative cos ๐‘ฅ plus ๐‘. So therefore, in our case, itโ€™s gonna be equal to negative one over seven cos ๐‘ข. And again, weโ€™ll add the constant of integration on at the end. Then finally, what we do is we put back in our substitution. And it takes us to negative one over seven cos seven ๐‘ฅ, which is exactly what we got using the general integral that we knew. So great. Iโ€™ve shown where we got our answer from using substitution. So now we can move on to the next part.

Now, if we integrate the right-hand side, weโ€™re gonna get four multiplied by. And then, if we integrate cos seven ๐‘ฅ, we get one over seven sin seven ๐‘ฅ. And we get that using the same method as the left-hand side. Except this time, the standard integral that weโ€™re using is the integral of cos ๐‘ฅ, which is equal to sin ๐‘ฅ plus ๐‘. And again, I couldโ€™ve shown this using substitution, which Iโ€™ve done on the left-hand side, where Iโ€™ve swapped in cos ๐‘ข instead of sin ๐‘ข and followed the process through. And once again, itโ€™s brought us to the right result, which in this case is one over seven sin seven ๐‘ฅ. So when we bring this all together, what we get is negative three over seven cos seven ๐‘ฅ plus four over seven sin seven ๐‘ฅ plus ๐‘. And we got that because if you multiply three by negative one over seven, you get negative three over seven. And if you multiply four by one over seven, you get four over seven.

And this brings us to our final answer, which Iโ€™ve got just by rearranging to have the positive term on the left, which is four over seven sin seven ๐‘ฅ minus three over seven cos seven ๐‘ฅ plus ๐‘.

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