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AP Calculus BC : Fundamental Theorem of Calculus with Definite Integrals

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Example Questions

Example Question #51 : Integrals

Evaluate the following integral

$\int_0^5 3x^4-12x^3+10x+12dx$

Possible Answers:

1875

185

685

625

Correct answer:

185

Explanation:

Evaluate the following integral

$\int_0^5 3x^4-12x^3+10x+12dx$

Let's begin by recalling our "reverse power rule" AKA, the antiderivative form of our power rule.

$\int x^ndx=\frac{x^{n+1}}{n+1}+c$

In other words, all we need to do for each term is increase the exponent by 1 and then divide by that number.

$\int_0^5 3x^4-12x^3+10x+12dx=\frac{3x^5}{5}-\frac{12x^4}{4}+\frac{10x^2}{2}+12x+c\mid_0^5$

Let's clean it up a little to get:

$\frac{3x^5}{5}-3x^4+5x^2+12x+c\mid_0^5$

Now, to evaluate our integral, we need to plug in 5 and 0 for x and find the difference between the values. In other words, if our integrated function is F(x), we need to find F(5)-F(0).

Let's start with F(5)

$\frac{3(5)^5}{5}-3(5)^4+5(5)^2+12(5)+c=1875-1875+125+60+c=185+c$

Next, let's look at F(0). If you look at our function carefully, you will notice that F(0) will cancel out all of our terms except for +c. So, we have the following:

F(5)=185+c

F(0)=c

Finding the difference cancels out the c's and leaves us with 185.

185+c-c=185

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AP Calculus BC : Fundamental Theorem of Calculus with Definite Integrals

Study concepts, example questions & explanations for AP Calculus BC

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Example Question #51 : Integrals

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