Video Transcript
Determine the integral of negative
three 𝑒 to the 𝜃 plus two tan 𝜃 sec 𝜃 𝑑𝜃.
In this question, we are asked to
integrate an expression with respect to 𝜃. To do this, we will split the
integrand so we are integrating term by term. We will begin by integrating
negative three 𝑒 to the 𝜃 with respect to 𝜃. We recall that the integral of 𝑒
to the power of 𝑥 with respect to 𝑥 is simply 𝑒 to the power of 𝑥 plus a
constant of integration 𝑐. This means that the integral of
negative three 𝑒 to the power of 𝜃 is simply negative three 𝑒 to the power of
𝜃. At this stage, we will not add a
constant of integration as we can just do this at the end of our expression.
Next, we need to consider the
integral of two tan 𝜃 sec 𝜃. One way of doing this is to first
recall that the derivative of sec 𝜃 with respect to 𝜃 is tan 𝜃 multiplied by sec
𝜃. This tells us that sec 𝜃 is an
antiderivative of tan 𝜃 sec 𝜃. And hence, the integral of tan 𝜃
sec 𝜃 with respect to 𝜃 is equal to sec 𝜃 plus a constant of integration 𝑐.
We notice that this expression
appears in our integrand. However, it has been multiplied by
two. The integral of two tan 𝜃 sec 𝜃
is therefore equal to two sec 𝜃. Adding a constant of integration
𝑐, we now have our final answer. The integral of negative three 𝑒
to the 𝜃 plus two tan 𝜃 sec 𝜃 with respect to 𝜃 is equal to negative three 𝑒 to
the 𝜃 plus two sec 𝜃 plus 𝑐.
