### Video Transcript

Determine the integral of the function negative seven multiplied by the cos of three 𝑥 multiplied by the tan of three 𝑥 with respect to 𝑥.

The first thing we should try in a problem like this is to see if our integral is in a standard form which we can integrate directly. We can see that our integrand is of the form of a cosine function multiplied by the tangent function, which is not a standard form which we know how to integrate. This means we’re going to need to manipulate this expression into a form which we can integrate.

There are multiple things we could try at this point. We could attempt to use the multiple angle formulas, or we could use the tangent identity to rewrite the function tan of three 𝑥. The tangent identity says that the tangent of 𝜃 is equivalent to the sin of 𝜃 divided by the cos of 𝜃. Since this identity is true for all values of 𝜃, we can replace 𝜃 with three 𝑥, giving us that the tan of three 𝑥 is equivalent to the sin of three 𝑥 divided by the cos of three 𝑥.

We can then replace the tan of three 𝑥 in our integrand with the sin of three 𝑥 divided by the cos of the three 𝑥. We can then simplify this by canceling the shared factor of the cos of three 𝑥, which gives us the integral of negative seven multiplied by the sin of three 𝑥 with respect to 𝑥. We know that for any constant 𝑘, the integral of 𝑘 multiplied by 𝑓 of 𝑥 with respect to 𝑥 is equal to 𝑘 multiplied by the integral of 𝑓 of 𝑥 with respect to 𝑥.

So, we can use this to take the constant of negative seven out of our integral. We can then use the fact that the integral of sin of 𝑎𝑥 with respect to 𝑥 is equal to negative the cos of 𝑎𝑥 divided by 𝑎 plus a constant of integration to evaluate our integral. Which gives us negative seven multiplied by negative the cos of three 𝑥 divided by three plus a constant of integration 𝐶 one. We can simplify this expression to give us seven multiplied by the cos of three 𝑥 over three minus seven times 𝐶 one.

Finally, we notice that negative seven multiplied by 𝐶 one is a constant, so we can replace this with a new constant, which we will call 𝐶. Giving us our final answer to the integral of negative seven multiplied by the cos of three 𝑥 multiplied by the tan of three 𝑥 with respect to 𝑥 is equal to seven multiplied by the cos of three 𝑥 divided by three plus a constant of integration 𝐶.