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AP Calculus BC : Fundamental Theorem of Calculus and Techniques of Antidifferentiation

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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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Example Question #1 : Improper Integrals

Evaluate:

$\int_{0}^{\infty } \frac{\cos x\; \mathrm{d}x}{e^{x}}$

Possible Answers:

$\frac{1}{2}$

$\frac{1}{2e}$

$\frac{\sqrt{2}}{2e}$

$\frac{1}{e}$

Correct answer:

$\frac{1}{2}$

Explanation:

First, we will find the indefinite integral $\int \frac{\cos x\; \mathrm{d}x}{e^{x}} = \int e^{-x} \cos x\; \mathrm{d}x$ using integration by parts.

We will let $u = \cos x$ and $\mathrm{d}v= e^{-x}\; \mathrm{d}x$ .

Then $\frac{\mathrm{d} u}{\mathrm{d} x} = - \sin x$ and $\mathrm{d} u = - \sin x \; \mathrm{d} x$ .

$v= \int e^{-x}\; \mathrm{d}x = - e^{-x} = - \frac{1}{e^{x} }$

$\int \frac{\cos x\; \mathrm{d}x}{e^{x}} = \int e^{-x} \cos x\; \mathrm{d}x$

$=\int u \; \mathrm{d} v$

$= uv - \int v \; \mathrm{d} u$

$= \cos x \cdot \left (- \frac{1}{e^{x} } \right ) - \int \left (- \frac{1}{e^{x} } \right ) \; \cdot \left (- \sin x \right )\; \mathrm{d} x$

$= - \frac{\cos x}{e^{x} } - \int e^{-x} \sin x \; \mathrm{d} x$

To find $\int e^{-x} \sin x \; \mathrm{d} x$ , we use another integration by parts:

$u = \sin x$ , which means that $\mathrm{d} u = \cos x \; \mathrm{d} x$ , and

$\mathrm{d}v= e^{-x}\; \mathrm{d}x$ , which means that, again, $v= - \frac{1}{e^{x} }$ .

$\int e^{-x} \sin x \; \mathrm{d} x$

$=\int u \; \mathrm{d} v$

$= uv - \int v \; \mathrm{d} u$

$= \sin x \cdot \left (- \frac{1}{e^{x} } \right ) - \int \left (- \frac{1}{e^{x} } \right ) \; \cdot \cos x \; \mathrm{d} x$

$= - \frac{ \sin x }{e^{x} } + \int \frac{\cos x \; \mathrm{d} x}{e^{x} }$

$\int \frac{\cos x\; \mathrm{d}x}{e^{x}} = - \frac{\cos x}{e^{x} } - \int e^{-x} \sin x \; \mathrm{d} x$

$\int \frac{\cos x\; \mathrm{d}x}{e^{x}} = - \frac{\cos x}{e^{x} } - \left ( - \frac{ \sin x }{e^{x} } + \int \frac{\cos x \; \mathrm{d} x}{e^{x} } \right )$

$\int \frac{\cos x\; \mathrm{d}x}{e^{x}} = - \frac{\cos x}{e^{x} } + \frac{ \sin x }{e^{x} } - \int \frac{\cos x \; \mathrm{d} x}{e^{x} }$

$2 \int \frac{\cos x\; \mathrm{d}x}{e^{x}} = - \frac{\cos x}{e^{x} } + \frac{ \sin x }{e^{x} }$

$2 \int \frac{\cos x\; \mathrm{d}x}{e^{x}} = \frac{\sin x -\cos x}{e^{x} }$

$\int \frac{\cos x\; \mathrm{d}x}{e^{x}} = \frac{\sin x -\cos x}{2e^{x} }$

Since

$\frac{-2}{2e^{x} } \leq \frac{\sin x -\cos x}{2e^{x} } \leq \frac{2}{2e^{x} }$ , or,

$- \frac{1}{e^{x} } \leq \frac{\sin x -\cos x}{2e^{x} } \leq \frac{1}{e^{x} }$

for all real , and

$- \lim_{x\rightarrow \infty } \frac{1}{e^{x} } = \lim_{x\rightarrow \infty } \frac{1}{e^{x} } = 0$ ,

by the Squeeze Theorem,

$\lim_{x\rightarrow \infty } \frac{\sin x -\cos x}{2e^{x} } = 0$ .

$\int_{0}^{\infty } \frac{\cos x\; \mathrm{d}x}{e^{x}}$

$= \lim_{b\rightarrow \infty }\int_{0}^{b } \frac{\cos x\; \mathrm{d}x}{e^{x}}$

$= \lim_{ b\rightarrow \infty }$ $\left.\begin{matrix} \frac{\sin x -\cos x}{2e^{x} } \\ \textrm{ } \end{matrix}\right|$ $\begin{matrix} b\\ \\ 0 \end{matrix}$

$= 0 - \frac{\sin 0 -\cos 0}{2e^{0} }$

$= - \frac{ 0 -1}{2 \cdot 1 } = \frac{1}{2}$

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Example Question #12 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Evaluate:

$\int_{0}^{1} x \ln x \; \mathrm{d}x$

Possible Answers:

$-\frac{1}{4}$

$\frac{1}{4}$

The integral does not converge

$\ln 4$

$-\ln 4$

Correct answer:

$-\frac{1}{4}$

Explanation:

First, we will find the indefinite integral, $\int x \ln x \; \mathrm{d}x$ .

We will let $u = \ln x$ and $\mathrm{d}v= x \; \mathrm{d}x$ .

Then,

$\frac{\mathrm{d} u }{\mathrm{d} x}= \frac{1}{x}$ and $\mathrm{d} u = \frac{\mathrm{d} x}{x}$ .

$v= \int x \; \mathrm{d}x =\frac{ x ^{2}}{2}$

and

$\int x \ln x \; \mathrm{d}x$

$=\int u \; \mathrm{d} v$

$= uv - \int v \; \mathrm{d} u$

$= \ln{x} \cdot \frac{ x ^{2}}{2} - \int \left (\frac{ x ^{2}}{2} \right ) \frac{\mathrm{d} x}{x}$

$= \frac{ x ^{2}\ln{x}}{2} - \frac{1}{2}\int x\; \mathrm{d} x$

$= \frac{ x ^{2}\ln{x}}{2} - \frac{1}{2} \cdot \frac{x^{2}}{2}$

$= \frac{ x ^{2}\ln{x}}{2} - \frac{x^{2}}{4}$

Now, this expression evaluated at x = 1 is equal to

$\frac{ 1 ^{2}\ln{1}}{2} - \frac{1^{2}}{4} = 0 - \frac{1}{4} = - \frac{1}{4}$ .

At x = 0 it is undefined, because $\ln 0$ does not exist.

We can use L'Hospital's rule to find its limit as $x\rightarrow 0$ , as follows:

$\lim_{x\rightarrow 0}\frac{2 x ^{2}\ln{x} - x^{2}}{4} =\lim_{x\rightarrow 0} \frac{2 \ln{x} - 1}{\frac{4}{x ^{2}}}$

$\lim_{x\rightarrow 0} \left (2 \ln{x} - 1 \right ) = - \infty$ and $\lim_{x\rightarrow 0} \frac{4}{x ^{2}} = \infty$ , so by L'Hospital's rule,

$\lim_{x\rightarrow 0} \frac{2 \ln{x} - 1}{\frac{4}{x ^{2}}}=\lim_{x\rightarrow 0} \frac{\frac {2}{x} }{\frac{-8}{x ^{3}}} = \lim_{x\rightarrow 0}\left ( -\frac{x^{2}}{4} \right ) = 0$

Therefore,

$\int_{0}^{1} x \ln x \; \mathrm{d}x$

$= \left.\begin{matrix} \frac{ x ^{2}\ln{x}}{2} - \frac{x^{2}}{4} \\ \\ \end{matrix}\right| \begin{matrix} 1\\ \\ 0 \end{matrix}$

$=\lim _{b\rightarrow 0}\left [ \left.\begin{matrix} \frac{ x ^{2}\ln{x}}{2} - \frac{x^{2}}{4} \\ \\ \end{matrix}\right| \begin{matrix} 1\\ \\ b \end{matrix} \right ]$

$= -\frac{1}{4} - 0$

$= -\frac{1}{4}$

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Example Question #1 : Improper Integrals

Evaluate:

$\int_{0 }^{\infty} \frac{x\; \mathrm{d}x}{e^{x^{2}}}$

Possible Answers:

$\frac{2}{e}$

$\frac{e}{2}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

Rewrite the integral as

$\int_{0 }^{\infty} \frac{x\; \mathrm{d}x}{e^{x^{2}}} = \int_{0 }^{\infty}e^{-x^{2}}\cdot x \; \mathrm{d}x} }$ .

Substitute $u = - x^{2}$ . Then

$\frac{\mathrm{d} u}{\mathrm{d} x} = - 2x$ and $-\frac{1}{2} \mathrm{d} u = x \; \mathrm{d} x$ . The lower bound of integration stays $u = - 0^{2} = 0$ , and the upper bound becomes $\lim_{x\rightarrow \infty } - x^{2} = - \infty$ , so

$\int_{0 }^{\infty} \frac{x\; \mathrm{d}x}{e^{x^{2}}} = \int_{0 }^{\infty}e^{-x^{2}}\cdot x \; \mathrm{d}x} }=-\frac{1}{2} \int_{0 }^{-\infty}e^{u} \mathrm{d}u}= \frac{1}{2} \int_{-\infty }^{0}e^{u} \mathrm{d}u}$

$=\frac{1}{2} \lim_{b\rightarrow -\infty } \left.\begin{matrix} e ^{u}\\ \textrm{ } \\ \textrm{ } \end{matrix}\right| \begin{matrix} 0\\ \\ b \end{matrix}$

Since $\lim_{b\rightarrow -\infty } e ^{u} = 0$ , the above is equal to

$=\frac{1}{2} (e^{0}-0) =\frac{1}{2} (1) = \frac{1}{2}$ .

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Example Question #1 : Improper Integrals

Evaluate $\int_{1}^{\infty}\frac{4}{x^{5}}dx$ .

Possible Answers:

$-\infty$

$\infty$

Correct answer:

Explanation:

By the Formula Rule, we know that $\int_{a}^{\infty}F(x)dx=\lim_{b\rightarrow \infty}\int_{a}^{b}f(x)dx$ . We therefore know that $\int_{1}^{\infty}\frac{4}{x^{5}}dx=\lim_{b\rightarrow \infty}\int_{1}^{b}\frac{4}{x^{5}}dx$ .

Continuing the calculation:

$\int_{1}^{b}\frac{4}{x^{5}}dx$

$=4\int_{1}^{b}\frac{1}{x^{5}}dx$

$=4\int_{1}^{b}{x^{-5}}dx$

By the Power Rule for Integrals, $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for all $n\neq-1$ with an arbitrary constant of integration . Therefore:

$4\int_{1}^{b}{x^{-5}}dx$

$=4({\frac{x^{-5+1}}{-5+1}})_{1}^{b}$

$=4(\frac{x^{-4}}{-4}})_{1}^{b}$

$=4(-{\frac{1}{4x^{4}}})_{1}^b$ .

So, $\lim_{b\rightarrow \infty }4(-{\frac{1}{4x^{4}}})_{1}^b$

$=\lim_{b\rightarrow \infty }4[-{\frac{1}{4b^{4}}-(-\frac{1}{4}})]$

$=\lim_{b\rightarrow \infty }4[-{\frac{1}{4b^{4}}+\frac{1}{4}})]$

$=\lim_{b\rightarrow \infty }[-{\frac{1}{b^{4}}+1}]$ .

=0+1

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Example Question #61 : Integrals

$\int\frac{x-1}{(x-3)(x+4)} dx=$

Possible Answers:

$\frac{(x-3)^2}{}7+\frac{2(x+4)^2}{7}+C$

$\frac{5}{7(x+4)}+\frac{2}{7(x+2)}+C$

$\frac{1}{7}\ln|x-2|+\frac{1}{7}\ln|x-4|+C$

$\frac{2}{7}\ln|x-3|+\frac{5}{7}\ln| x+4|+C$

$\frac{5}{7(x+4)}\ln |x|+\frac{2}{7(x-3)}\ln |x|+C$

Correct answer:

$\frac{2}{7}\ln|x-3|+\frac{5}{7}\ln| x+4|+C$

Explanation:

In order to evaluate this integral, we will need to use partial fraction decomposition.

$\frac{x-1}{(x-3)(x+4)}=\frac{A}{x-3}+\frac{B}{x+4}$

Multiply both sides of the equation by the common denominator, which is (x-3)(x+4).

x-1=A(x+4)+B(x-3)

x-1=Ax+4A+Bx-3B=(A+B)x+4A-3B

This means that A+B must equal 1, and 4A-3B=-1.

A+B=1

A=1-B

4(1-B)-3B=-1

B=5/7, A=2/7

$\frac{x-1}{(x-3)(x+4)}=\frac{2/7}{x-3}+\frac{5/7}{x+4}$

$\int(\frac{2}{7}\cdot \frac{1}{x-3}+\frac{5}{7}\cdot \frac{1}{x+4})dx$

$=\int\frac{2}{7}\cdot\frac{1}{x-3} dx +\int\frac{5}{7}\cdot\frac{1}{x+4}dx$

$=\frac{2}{7}\int\frac{dx}{x-3}+\frac{5}{7}\int\frac{dx}{x+4}$

$=\frac{2}{7}\ln|x-3|+\frac{5}{7}\ln| x+4|+C$

The answer is $\frac{2}{7}\ln|x-3|+\frac{5}{7}\ln| x+4|+C$ .

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Example Question #2 : Substitution Of Variables (By Parts And Simple Partial Fractions)

Integrate:

$\int \frac{dx}{2x^2+16}$

Possible Answers:

$\arccos(\frac{x}{4})+C$

$\frac{1}{\sqrt{2}}\arctan(\frac{\sqrt{2}x}{4})+C$

$\frac{1}{4\sqrt{2}}\arctan(\frac{x}{4})+C$

$\frac{1}{4\sqrt{2}}\arcsin(\frac{x}{4})+C$

$\frac{1}{4\sqrt{2}}\arctan(\frac{\sqrt{2}x}{4})+C$

Correct answer:

$\frac{1}{4\sqrt{2}}\arctan(\frac{\sqrt{2}x}{4})+C$

Explanation:

To integrate, we must first make the following substitution:

$u=\sqrt{2}x, du=\sqrt{2}dx$

Rewriting the integral in terms of u and integrating, we get

$\frac{1}{\sqrt{2}}\int \frac{du}{u^2+16}= \frac{1}{4\sqrt{2}}\arctan(\frac{u}{4})+C$

The following rule was used to integrate:

$\int \frac{du}{u^2+a^2}=\frac{1}{a}\arctan(\frac{x}{a})+C$

Finally, we replace u with our original x term:

$\frac{1}{4\sqrt{2}}\arctan(\frac{\sqrt{2}x}{4})+C$

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AP Calculus BC : Fundamental Theorem of Calculus and Techniques of Antidifferentiation

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #51 : Integrals

Example Question #1 : Improper Integrals

Example Question #12 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Example Question #1 : Improper Integrals

Example Question #1 : Improper Integrals

Example Question #61 : Integrals

Example Question #2 : Substitution Of Variables (By Parts And Simple Partial Fractions)

All AP Calculus BC Resources

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