Video: Integrating Trigonometric Functions

Determine ∫−5 tan 6𝑥 d𝑥.

02:42

Video Transcript

Determine the indefinite integral of negative five tan of six 𝑥 with respect to 𝑥.

In order to answer this question, recall the identity tan of 𝑥 equals sin 𝑥 over cos 𝑥. Replacing 𝑥 by six 𝑥 in this identity, we obtain that tan of six 𝑥 equals sin of six 𝑥 over cos of six 𝑥. Therefore, we can rewrite the integral in question as the indefinite integral of negative five sin of six 𝑥 over cos of six 𝑥 with respect to 𝑥.

Now, recall the property that given a nonzero function 𝑓, for all constants 𝑎, the indefinite integral of 𝑎 timesed by the derivative of 𝑓 with respect to 𝑥 𝑓 prime of 𝑥 over 𝑓 of 𝑥 with respect to 𝑥 is equal to 𝑎 timesed by the natural logarithm of the absolute value of 𝑓 of 𝑥 plus 𝐶, where 𝐶 is the constant of integration. We are going to use this result to determine the integral given to us in the question.

Let 𝑓 of 𝑥 equal cos of six 𝑥. Then, using the fact that for all constants 𝑎, the derivative of the function cos of 𝑎𝑥 with respect to 𝑥 is negative 𝑎 sin of 𝑎𝑥. We find that the derivative of 𝑓 with respect to 𝑥 𝑓 prime of 𝑥 is equal to negative six sin of six 𝑥. Dividing both sides of this equation by six and then multiplying both sides of the equation by five, we obtain that five over six multiplied by 𝑓 prime of 𝑥 equals negative five sin of six 𝑥. Therefore, substituting this and 𝑓 of 𝑥 equals cos of six 𝑥 into the integral in question, we can rewrite the integral as the indefinite integral of five over six multiplied by 𝑓 prime of 𝑥 over 𝑓 of 𝑥 with respect to 𝑥.

Now, letting 𝑎 equal five over six in the property mentioned earlier, we see that this integral evaluates to five over six multiplied by the natural logarithm of the absolute value of 𝑓 of 𝑥 plus 𝐶, where 𝐶 is the constant of integration. Substituting 𝑓 of 𝑥 equals cos of six 𝑥 into this, we obtain five over six multiplied by the natural logarithm of the absolute value of the function cos of six 𝑥 plus 𝐶, where 𝐶 is the constant of integration. This is our final answer.

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