Video: AP Calculus AB Exam 1 β€’ Section I β€’ Part A β€’ Question 3

Find d/dπ‘₯ ∫_(5)^(π‘₯⁴) ln(7 + 𝑑³) d𝑑.

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Video Transcript

Find d by dπ‘₯ of the integral evaluated between five and π‘₯ to the fourth power of the natural log of seven plus 𝑑 cubed with respect to 𝑑.

Now, as tempting as it may be, we absolutely do not want to evaluate the integral here. Instead, weβ€²re going to recall the second fundamental theorem of calculus. This says that if 𝑓 is a continuous function and π‘Ž is a constant, then the integral evaluated between π‘Ž and π‘₯ of 𝑓 of 𝑑 with respect to 𝑑 is equal to capital 𝐹 of π‘₯ minus capital 𝐹 of π‘Ž, where capital 𝐹 prime of π‘₯ is equal to the function of π‘₯. And remember, capital 𝐹 is the antiderivative of the function. So how does this help us?

Weβ€²re going to differentiate both sides of the equation with respect to π‘₯. And we see that we can separate our derivative into d by dπ‘₯ of capital 𝐹 of π‘₯ minus d by dπ‘₯ of capital 𝐹 of π‘Ž. Remember, capital 𝐹 of π‘Ž will give us a constant. This means itβ€²s independent of π‘₯. And so, its derivative is going to be zero. So we’re left with the derivative of capital 𝐹 of π‘₯ with respect to π‘₯ or capital 𝐹 prime of π‘₯. Earlier though, we said that capital 𝐹 prime of π‘₯ was equal to 𝑓 of π‘₯. And we therefore see that the derivative with respect to π‘₯ of the integral evaluated between π‘Ž and π‘₯ of 𝑓 of 𝑑 with respect to 𝑑 is equal to 𝑓 of π‘₯.

Letβ€²s see how we can use this to evaluate the derivative with respect to π‘₯ of the integral between five and π‘₯ to the fourth power of the natural log of seven plus 𝑑 cubed with respect to 𝑑. We can say that in our example, 𝑓 of 𝑑 is equal to the natural log of seven plus 𝑑 cubed. And we use the rules we developed earlier. And we see that the derivative with respect to π‘₯ is equal to d by dπ‘₯ of capital 𝐹 of π‘₯ to the fourth power minus capital 𝐹 of five. We split our derivative up. And then we recall that the derivative of capital 𝐹 of five with respect to π‘₯ is going to be equal to zero, since itβ€²s going to be independent of π‘₯.

And so, we see that the derivative with respect to π‘₯ of the integral evaluated between five and π‘₯ to the fourth power of the natural log of seven plus 𝑑 cubed with respect to 𝑑 is the derivative of capital 𝐹 of π‘₯ to the fourth power with respect to π‘₯. Weβ€²re going to need to be really careful here when evaluating d by dπ‘₯ of capital 𝐹 of π‘₯ to the power of four. Weβ€²re going to be differentiating a function of a function, or a composite function. So weβ€²re going to need to use the chain rule.

This says that if 𝑦 is a function in 𝑒 and 𝑒 itself is a function in π‘₯, then the derivative of 𝑦 with respect to π‘₯ is equal to d𝑦 by d𝑒 times d𝑒 by dπ‘₯. So this means that the derivative of capital 𝐹 of π‘₯ to the power of four with respect to π‘₯ is equal to the derivative of capital 𝐹 of π‘₯ to the power of four, which is just 𝐹 prime of π‘₯ to the power of four multiplied by the derivative of π‘₯ to the power of four with respect to π‘₯. That’s four π‘₯ cubed.

Now, we said earlier that 𝐹 prime of π‘₯ was equal to 𝑓 of π‘₯. So we can say that 𝐹 prime of π‘₯ to the power of four is equal to 𝑓 of π‘₯ to the power of four. Since we said that 𝑓 of 𝑑 was equal to the natural log of seven plus 𝑑 cubed, we know that 𝑓 of π‘₯ to the fourth power is equal to the natural log of seven plus π‘₯ to the fourth power cubed times four π‘₯ cubed. π‘₯ to the fourth power cubed is π‘₯ to the power of 12. And we can see that the derivative with respect to π‘₯ of the integral evaluated between five and π‘₯ to the fourth power of the natural log of seven plus 𝑑 cubed with respect to 𝑑 is equal to four π‘₯ cubed times the natural log of seven plus π‘₯ to the power of 12.

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