Video Transcript
Determine the integral of three sin
seven ๐ฅ plus four cos seven ๐ฅ d๐ฅ.
So the first thing you can do is
split our expression. And thatโs because if weโre gonna
integrate it, we can integrate each term separately. So weโve got the integral of three
sin seven ๐ฅ d๐ฅ plus the integral of four cos seven ๐ฅ d๐ฅ. 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 ๐ฅ d๐ฅ plus four multiplied by the integral
of cos seven ๐ฅ d๐ฅ. And we can do that because they
wonโt affect the integration itself. So what Iโm gonna do is deal with
the left-hand side first. And what Iโm gonna do is integrate
sin seven ๐ฅ. And when I integrate sin seven ๐ฅ,
I get negative one over seven cos seven ๐ฅ. And now, Iโd have plus ๐. And thatโs the constant of
integration. But Iโm gonna add that in at the
end.
Now, the reason I got that is
because we know that the integral of sin ๐๐ฅ d๐ฅ is equal to negative one over ๐
cos ๐๐ฅ plus ๐. In our case, the ๐ 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 ๐ข is equal to seven ๐ฅ. So then, what we can do is
differentiate to find d๐ข d๐ฅ. And when we do that, we get d๐ข d๐ฅ
is equal to seven. And thatโs because if we
differentiate seven ๐ฅ, we just get seven if we differentiate with respect to
๐ฅ. And then, we can say that d๐ฅ is
equal to one over seven d๐ข. And we get that by dealing with d๐ข
d๐ฅ 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 weโre gonna get is
one-seventh, cause Iโve taken the constant outside the integration, multiplied by
the integral of sin ๐ข d๐ข. And we know the integral of sin ๐ข,
because the integral of sin ๐ฅ is one of our standard integrals. And the integral of sin ๐ฅ is equal
to negative cos ๐ฅ plus ๐. So therefore, in our case, itโs
gonna be equal to negative one over seven cos ๐ข. And again, weโll 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 ๐ฅ, which is exactly what we got using the general integral
that we knew. So great. Iโve 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, weโre gonna get four multiplied by. And then, if we integrate cos seven
๐ฅ, we get one over seven sin seven ๐ฅ. And we get that using the same
method as the left-hand side. Except this time, the standard
integral that weโre using is the integral of cos ๐ฅ, which is equal to sin ๐ฅ plus
๐. And again, I couldโve shown this
using substitution, which Iโve done on the left-hand side, where Iโve swapped in cos
๐ข instead of sin ๐ข and followed the process through. And once again, itโs brought us to
the right result, which in this case is one over seven sin seven ๐ฅ. So when we bring this all together,
what we get is negative three over seven cos seven ๐ฅ plus four over seven sin seven
๐ฅ 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 Iโve got just by rearranging to have the positive term on the left,
which is four over seven sin seven ๐ฅ minus three over seven cos seven ๐ฅ plus
๐.
