Video Transcript
Determine the integral of three sin
seven 𝑥 plus four cos seven 𝑥 d𝑥.
So the first thing you can do is
split our expression. And that’s because if we’re gonna
integrate it, we can integrate each term separately. So we’ve got the integral of three
sin seven 𝑥 d𝑥 plus the integral of four cos seven 𝑥 d𝑥. 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 𝑥 d𝑥 plus four multiplied by the integral
of cos seven 𝑥 d𝑥. And we can do that because they
won’t affect the integration itself. So what I’m gonna do is deal with
the left-hand side first. And what I’m gonna do is integrate
sin seven 𝑥. And when I integrate sin seven 𝑥,
I get negative one over seven cos seven 𝑥. And now, I’d have plus 𝑐. And that’s the constant of
integration. But I’m gonna add that in at the
end.
Now, the reason I got that is
because we know that the integral of sin 𝑎𝑥 d𝑥 is equal to negative one over 𝑎
cos 𝑎𝑥 plus 𝑐. In our case, the 𝑎 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 𝑢 is equal to seven 𝑥. So then, what we can do is
differentiate to find d𝑢 d𝑥. And when we do that, we get d𝑢 d𝑥
is equal to seven. And that’s because if we
differentiate seven 𝑥, we just get seven if we differentiate with respect to
𝑥. And then, we can say that d𝑥 is
equal to one over seven d𝑢. And we get that by dealing with d𝑢
d𝑥 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 we’re gonna get is
one-seventh, cause I’ve taken the constant outside the integration, multiplied by
the integral of sin 𝑢 d𝑢. And we know the integral of sin 𝑢,
because the integral of sin 𝑥 is one of our standard integrals. And the integral of sin 𝑥 is equal
to negative cos 𝑥 plus 𝑐. So therefore, in our case, it’s
gonna be equal to negative one over seven cos 𝑢. And again, we’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 𝑥, which is exactly what we got using the general integral
that we knew. So great. I’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, we’re gonna get four multiplied by. And then, if we integrate cos seven
𝑥, we get one over seven sin seven 𝑥. And we get that using the same
method as the left-hand side. Except this time, the standard
integral that we’re using is the integral of cos 𝑥, which is equal to sin 𝑥 plus
𝑐. And again, I could’ve shown this
using substitution, which I’ve done on the left-hand side, where I’ve swapped in cos
𝑢 instead of sin 𝑢 and followed the process through. And once again, it’s brought us to
the right result, which in this case is one over seven sin seven 𝑥. So when we bring this all together,
what we get is negative three over seven cos seven 𝑥 plus four over seven sin seven
𝑥 plus 𝑐. 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 I’ve got just by rearranging to have the positive term on the left,
which is four over seven sin seven 𝑥 minus three over seven cos seven 𝑥 plus
𝑐.
