# Question Video: Finding the General Antiderivative of a Given Function Involving Trigonometric and Exponential Functions Mathematics • Higher Education

Determine ∫ −3𝑒^𝜃 + 2 tan 𝜃 sec 𝜃 𝑑𝜃.

02:19

### 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.

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