Video Transcript
Determine the integral of six times the sin of seven 𝑥 plus five multiplied by the sec squared of two 𝑥 with respect to 𝑥.
We’re asked to evaluate the integral of the sum of two trigonometric functions, and we know how to evaluate the integral of both of these. So we’ll evaluate our integral term by term. We have the integral of six times the sin of seven 𝑥 with respect to 𝑥 plus the integral of five times the sec squared of two 𝑥 with respect to 𝑥. And we can just evaluate each of these integrals separately. Let’s start with our first integral.
We need to recall the following standard trigonometric integral result. For any real constants 𝑎 and 𝑘, where 𝑎 is not equal to zero, the integral of 𝑘 times the sin of 𝑎𝑥 with respect to 𝑥 is equal to negative 𝑘 over 𝑎 times the cos of 𝑎𝑥 plus our constant of integration 𝐶. And we can apply this to directly evaluate our integral. We can see the value of 𝑘 is six and the value of 𝑎 is seven. So by applying our integral result, with 𝑘 equal to six and 𝑎 equal to seven, we get negative six divided by seven times the cos of seven 𝑥 plus our constant of integration we’ll call capital 𝐴.
We can now move on to evaluating our second integral. Once again, we’re going to need to recall a standard trigonometric integral result. For any real constants 𝑎 and 𝑘, where 𝑎 is not equal to zero, the integral of 𝑘 times the sec squared of 𝑎𝑥 with respect to 𝑥 is equal to 𝑘 over 𝑎 times the tan of 𝑎𝑥 plus our constant of integration 𝐶. This time, we can see our value of the constant 𝑘 is equal to five and our value of the constant 𝑎 is equal to two. So by substituting 𝑎 is equal to two and 𝑘 is equal to five into this expression, we get five over two times the tan of two 𝑥 plus our constant of integration we’ll call 𝐵.
And we could leave our answer like this. However, we can simplify even further. We notice that both 𝐴 and 𝐵 are constants. So we can introduce a new constant called 𝐶 which is equal to their sum, 𝐴 plus 𝐵. And doing this gives us our final answer. Therefore, we were able to show the integral of six times the sin of seven 𝑥 plus five times the sec squared of two 𝑥 with respect to 𝑥 is equal to negative six over seven times the cos of seven 𝑥 plus five over two multiplied by the tan of two 𝑥 plus our constant of integration 𝐶.
