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AP Calculus AB : Antiderivatives following directly from derivatives of basic functions

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Example Questions

Example Question #41 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Integrate:

$\int_{0}^{4}(x-x^2)dx$

Possible Answers:

$\frac{88}{3}$

$\frac{40}{3}$

$-\frac{40}{3}$

Correct answer:

$-\frac{40}{3}$

Explanation:

The integral is equal to

$\int_{0}^{4}(x-x^2)dx = \frac{x^2}{2}-\frac{x^3}{3}\left.\begin{matrix} 4\\0 \end{matrix}\right|$

The following rule was used to integrate:

$\int x^n dx=\frac{x^{n+1}}{n+1}+C$

Now, we find the numerical answer by plugging in the upper bound of integration and subtracting what we get from plugging in the lower bound of integration:

$8-\frac{64}{3}-(0-0)=-\frac{40}{3}$

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Example Question #41 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Integrate:

$\int_{0}^{2\pi}5\sec^2(x)dx$

Possible Answers:

Undefined

$\pi$

Correct answer:

Explanation:

The integral is equal to

$\int_{0}^{2\pi}5\sec^2(x)dx=5\tan(x)\left.\begin{matrix} 2\pi\\0 \end{matrix}\right|$

and was found using the identical rule.

Evaluating by plugging in the upper bound and subtracting from what we get from plugging in the lower bound, we get

$5(0)-(5 \cdot 0)=0$

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AP Calculus AB : Antiderivatives following directly from derivatives of basic functions

Study concepts, example questions & explanations for AP Calculus AB

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Example Question #41 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Example Question #41 : Antiderivatives Following Directly From Derivatives Of Basic Functions

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