Video Transcript
Determine the integral of seven sec
𝑥 multiplied by tan 𝑥 minus five sec 𝑥 with respect to 𝑥.
In this question, since there is a
factored expression within the integrand, we will start by expanding through the
parentheses in the integrand. This gives us seven sec 𝑥 tan 𝑥
minus 35 sec squared 𝑥. And it is this expression we need
to integrate with respect to 𝑥. The first term is the product of
the secant and tangent function, and the second term is the square of the secant
function.
In order to solve our problem, we
recall the following definite integrals. Firstly, the integral of sec 𝑥 tan
𝑥 with respect to 𝑥 is equal to sec 𝑥 plus the constant of integration 𝐶. And the integral of sec squared 𝑥
with respect to 𝑥 is equal to tan 𝑥 plus 𝐶. We can factor out the constants and
separate our integrand as shown. We have seven multiplied by the
integral of sec 𝑥 tan 𝑥 with respect to 𝑥 minus 35 multiplied by the integral of
sec squared 𝑥 with respect to 𝑥.
Applying the two formulae, we
obtain seven multiplied by sec 𝑥 plus 𝐶 one minus 35 multiplied by tan 𝑥 plus 𝐶
two, for some arbitrary constant 𝐶 one and 𝐶 two. Since after distributing through
the parentheses, we will end up with a combination of 𝐶 one and 𝐶 two, we can
replace this expression by another arbitrary constant 𝐶 to write the solution as
seven sec 𝑥 minus 35 tan 𝑥 plus 𝐶. This is the integral of seven sec
𝑥 multiplied by tan 𝑥 minus five sec 𝑥 with respect to 𝑥.
