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

Determine ∫ (27 sin π‘₯ + 21 cos π‘₯)/(7 sin π‘₯ βˆ’ 9 cos π‘₯) dπ‘₯.

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Video Transcript

Determine the indefinite integral of 27 sin π‘₯ plus 21 cos π‘₯ all over seven sin π‘₯ minus nine cos π‘₯ with respect to π‘₯.

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 π‘₯ with respect to π‘₯ is equal to cos π‘₯ and the differential of negative cos π‘₯ with respect to π‘₯ is equal to sin π‘₯. 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 𝑒 be equal to the denominator of the integrand, so that seven sin π‘₯ minus nine cos π‘₯. Now we can differentiate 𝑒 with respect to π‘₯. Using the fact that sine differentiates to cos π‘₯ and negative cos π‘₯ differentiates to sin π‘₯. We obtain that d𝑒 by dπ‘₯ is equal to seven cos π‘₯ plus nine sin π‘₯. This gives us that d𝑒 is equal to nine sin π‘₯ plus seven cos π‘₯ dπ‘₯.

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 π‘₯ minus nine cos π‘₯ multiplied by nine sin π‘₯ plus seven cos π‘₯ dπ‘₯. And so we can substitute 𝑒 into the denominator of our fraction. And we can substitute in d𝑒 for nine sin π‘₯ plus seven cos π‘₯ dπ‘₯. Giving us that it is equal to the integral of three over 𝑒 d𝑒, which can be integrated to three multiplied by the natural logarithm of the absolute value of 𝑒 plus 𝑐. For our final step, we simply substitute back in seven sin π‘₯ minus nine cos π‘₯ for 𝑒. This gives us our solution, which is three multiplied by the natural logarithm of the absolute value of seven sin π‘₯ minus nine cos π‘₯ plus 𝑐.

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