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

Determine โˆซ (2๐‘ฅ + secยฒ 6๐‘ฅ) d๐‘ฅ.

02:07

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‘Ž is not equal to zero, then the integral of the sec squared of ๐‘Ž๐‘ฅ with respect to ๐‘ฅ is equal to the tan of ๐‘Ž๐‘ฅ divided by ๐‘Ž 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‘Ž is equal to six. So, we set our value of ๐‘Ž 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.

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