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

Determine β« β3π^π + 2 tan π sec π ππ.

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### Video Transcript

Determine the integral of negative three π to the π plus two tan π sec π ππ.

In this question, we are asked to integrate an expression with respect to π. To do this, we will split the integrand so we are integrating term by term. We will begin by integrating negative three π to the π with respect to π. We recall that the integral of π to the power of π₯ with respect to π₯ is simply π to the power of π₯ plus a constant of integration π. This means that the integral of negative three π to the power of π is simply negative three π to the power of π. 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 π sec π. One way of doing this is to first recall that the derivative of sec π with respect to π is tan π multiplied by sec π. This tells us that sec π is an antiderivative of tan π sec π. And hence, the integral of tan π sec π with respect to π is equal to sec π plus a constant of integration π.

We notice that this expression appears in our integrand. However, it has been multiplied by two. The integral of two tan π sec π is therefore equal to two sec π. Adding a constant of integration π, we now have our final answer. The integral of negative three π to the π plus two tan π sec π with respect to π is equal to negative three π to the π plus two sec π plus π.