# Video: Finding the Integration of a Function Involving Trigonometric Functions Using Integration by Substitution

Determine ∫ (27 sin 𝑥 + 21 cos 𝑥)/(7 sin 𝑥 − 9 cos 𝑥) d𝑥.

01:50

### 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.