Question Video: Finding the Integration of a Function Involving Trigonometric and Logarithmic Functions Using Integration by Substitution Mathematics

Video Transcript

Determine the integral of four plus five times the tan of 𝑥 all divided by four 𝑥 plus five times the natural logarithm of eight multiplied by the sec of 𝑥 with respect to 𝑥.

In this question, we’re asked to find the indefinite integral of the quotient of two functions. And this is a very complicated-looking integrand. There’s a few different methods we could use to try and simplify this integrand. However, as we’ll see, the easiest way to evaluate this integral is to use one of our integral rules.

We can recall the integral of 𝑓 prime of 𝑥 divided by 𝑓 of 𝑥 with respect to 𝑥 is equal to the natural logarithm of the absolute value of 𝑓 of 𝑥 plus a constant of integration 𝐶. And this is a useful result for helping us evaluate the integral of the quotient of two functions, since we can just check whether the derivative of the denominator with respect to 𝑥 appears in the numerator. So let’s try and evaluate our integral by using this integral result.

To do this, we need to set 𝑓 of 𝑥 equal to the denominator of our integrand. We now want to find an expression for 𝑓 prime of 𝑥 to see if this appears in the numerator of our integrand. Since we would differentiate 𝑓 of 𝑥 term by term, we can already see the derivative of four 𝑥 with respect to 𝑥 is the coefficient of 𝑥, which is four. So we only need to find the derivative of the second term in our denominator to see if this is five times the tan of 𝑥.

And since this second term in the denominator is a composition of two functions, we’ll need to differentiate this by using the chain rule. We recall the chain rule tells us the derivative of 𝑢 evaluated at 𝑣 of 𝑥 with respect to 𝑥 is equal to 𝑣 prime of 𝑥 times 𝑢 prime evaluated at 𝑣 of 𝑥. We want to use this to differentiate five times the natural logarithm of eight times sec of 𝑥. So we’ll set 𝑣 of 𝑥 to be our inner function, that’s eight sec of 𝑥, and 𝑢 evaluated 𝑣 to be our outer function. That’s five times the natural logarithm of 𝑣.

Now, to apply the chain rule, we need to find expressions for 𝑢 prime of 𝑣 and 𝑣 prime of 𝑥. Let’s start by finding 𝑢 prime of 𝑣. To do this, we recall the derivative of the natural logarithm of 𝑣 with respect to 𝑣 is the reciprocal function, one over 𝑣. Therefore, 𝑢 prime of 𝑣 is the derivative of five times the natural logarithm of 𝑣, which is just five divided by 𝑣. Let’s now find 𝑣 prime of 𝑥. That’s the derivative of eight times the sec of 𝑥 with respect to 𝑥.

We can do this by recalling the derivative of the sec of 𝑥 with respect to 𝑥 is sec of 𝑥 multiplied by tan of 𝑥. Therefore, 𝑣 prime of 𝑥 is equal to eight times the sec of 𝑥 multiplied by the tan of 𝑥. We can now substitute our expressions for 𝑣 prime of 𝑥, 𝑢 prime, and 𝑣 of 𝑥 into the chain rule to find the derivative of five times the natural logarithms of eight sec of 𝑥. This gives us eight times the sec of 𝑥 multiplied by the tan of 𝑥 times five divided by eight sec of 𝑥.

And we can simplify this expression by canceling the shared factor of eight sec of 𝑥 in the numerator and denominator. This just leaves us with five tan of 𝑥. Therefore, the derivative of the second term in our denominator with respect to 𝑥 is the second term in the numerator. And in turn, this means the derivative of the denominator with respect to 𝑥 is equal to the numerator. So we can just use our integral rule to evaluate this integral. And this gives us our final answer. The integral of four plus five times the tan of 𝑥 all divided by four 𝑥 plus five natural logarithm of eight sec of 𝑥 with respect to 𝑥 is equal to the natural logarithm of the absolute value of four 𝑥 plus five times the natural logarithm of eight sec of 𝑥 plus a constant of integration 𝐶.