# Question Video: Integrating Trigonometric Functions Involving Reciprocal Trigonometric Functions Mathematics • Higher Education

Determine โซ (2๐ฅ + secยฒ 6๐ฅ) d๐ฅ.

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### Video Transcript

Determine the integral of two ๐ฅ plus the sec squared of six ๐ฅ with respect to ๐ฅ.

In this question, weโre asked to evaluate the integral of the sum of two functions, and we know how to integrate each of these two terms separately. So, the first thing weโll do is split our integral into two. Weโll split it into the integral of two ๐ฅ with respect to ๐ฅ plus the integral of the sec squared of six ๐ฅ with respect to ๐ฅ. And we can evaluate each of these integrals directly. Since our first term is a linear function, we can integrate this by using the power rule for integration.

We want to add one to our exponent of ๐ฅ and then divide by this new exponent. And to do this, we need to recall that ๐ฅ is equal to ๐ฅ to the first power. So, our exponent of ๐ฅ is equal to one. So, weโll add one to our exponent of one, giving us a new exponent of two, and then divide by this new exponent, giving us two ๐ฅ squared divided by two. And itโs worth pointing out here we donโt need to add our constant of integration because weโll get another constant of integration in our second integral. And we can just combine these two constants into one. And we can simplify two ๐ฅ squared divided by two to give us ๐ฅ squared.

Now, we want to evaluate the integral of the sec squared of six ๐ฅ with respect to ๐ฅ. And thereโs two different ways we could do this. We could recall the derivative of the tan of six ๐ฅ with respect to ๐ฅ is six times the sec squared of six ๐ฅ. This would then tell us the tan of six ๐ฅ all divided by six is an antiderivative of our integral. However, we already did this in the general case. We found if ๐ is not equal to zero, then the integral of the sec squared of ๐๐ฅ with respect to ๐ฅ is equal to the tan of ๐๐ฅ divided by ๐ plus our constant of integration ๐ถ. This is a standard trigonometric integral result which comes up a lot. Itโs worth committing this to memory.

In our case, the value of ๐ is equal to six. So, we set our value of ๐ equal to six. And weโll write the result as one six times the tangent of six ๐ฅ plus our constant of integration ๐ถ. And this gives us our final answer.

Therefore, we were able to show the integral of two ๐ฅ plus the sec squared of six ๐ฅ with respect to ๐ฅ is equal to ๐ฅ squared plus one-sixth times the tan of six ๐ฅ plus our constant of integration ๐ถ.