# Video: APCALC04AB-P1A-Q17-389197308614

The table shows some values of a differentiable function 𝑓 and its derivative 𝑓′. Find ∫_(1)^(4) 𝑓′(𝑥) d𝑥.

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### Video Transcript

The table shows some values of a differentiable function 𝑓 and its derivative 𝑓 prime. Find the definite integral between one and four of 𝑓 prime of 𝑥 with respect to 𝑥.

To answer this question, we’re going to recall the second part of the fundamental theorem of calculus. This says that if 𝑓 is a real-valued function on the closed interval 𝑎 to 𝑏 such that capital 𝐹 is an antiderivative of 𝑓 on this same interval. Then if 𝑓 is Reimann integrable on that closed interval, then the definite integral between 𝑎 and 𝑏 of 𝑓 of 𝑥 with respect to 𝑥 is equal to capital 𝐹 of 𝑏 minus capital 𝐹 of 𝑎.

We can see that 𝑓 of 𝑥 and 𝑓 prime of 𝑥 are real-valued functions on the closed interval one to four. Now, it also follows that if 𝑓 prime of 𝑥 is the derivative of 𝑓 of 𝑥, then 𝑓 of 𝑥 must itself be the antiderivative of 𝑓 prime of 𝑥. We can use this to rewrite our theorem such that the definite integral between 𝑎 and 𝑏 of 𝑓 prime of 𝑥 with respect to 𝑥 must be equal to 𝑓 of 𝑏 minus 𝑓 of 𝑎. In our case, this means that the definite integral between one and four of 𝑓 prime of 𝑥 with respect to 𝑥 must be 𝑓 of four minus 𝑓 of one. We can see from our table that 𝑓 of four is nine and 𝑓 of one is two.

So the definite integral between one and four of 𝑓 prime of 𝑥 with respect to 𝑥 is nine minus two, which is equal to seven.