Video Transcript
Determine the indefinite integral
of 27 sin 𝑥 plus 21 cos 𝑥 all over seven sin 𝑥 minus nine cos 𝑥 with respect to
𝑥.
Here, we notice that the numerator
of the integrand looks as though it could be the differential of the
denominator. Since the differential of sin 𝑥
with respect to 𝑥 is equal to cos 𝑥 and the differential of negative cos 𝑥 with
respect to 𝑥 is equal to sin 𝑥. Now it looks as though the
numerator may differ from the differential of the denominator by a constant
factor. However, we do not know what this
factor is. We can try and find it by using a
substitution. Let’s let 𝑢 be equal to the
denominator of the integrand, so that seven sin 𝑥 minus nine cos 𝑥. Now we can differentiate 𝑢 with
respect to 𝑥. Using the fact that sine
differentiates to cos 𝑥 and negative cos 𝑥 differentiates to sin 𝑥. We obtain that d𝑢 by d𝑥 is equal
to seven cos 𝑥 plus nine sin 𝑥. This gives us that d𝑢 is equal to
nine sin 𝑥 plus seven cos 𝑥 d𝑥.
Now let’s rearrange our integral so
we can apply this substitution. We notice that we can factor out a
factor of three from our numerator. And this enables us to write our
integral as the integral of three over seven sin 𝑥 minus nine cos 𝑥 multiplied by
nine sin 𝑥 plus seven cos 𝑥 d𝑥. And so we can substitute 𝑢 into
the denominator of our fraction. And we can substitute in d𝑢 for
nine sin 𝑥 plus seven cos 𝑥 d𝑥. Giving us that it is equal to the
integral of three over 𝑢 d𝑢, which can be integrated to three multiplied by the
natural logarithm of the absolute value of 𝑢 plus 𝑐. For our final step, we simply
substitute back in seven sin 𝑥 minus nine cos 𝑥 for 𝑢. This gives us our solution, which
is three multiplied by the natural logarithm of the absolute value of seven sin 𝑥
minus nine cos 𝑥 plus 𝑐.
