Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #442 : Finding Integrals

\displaystyle \int \frac{5x^2+20x-10}{5}dx

Possible Answers:

\displaystyle \frac{x^3}{3}-x^2-2x+C

\displaystyle x^3+2x^2-2x+C

\displaystyle \frac{x^3}{3}+2x^2-2x+C

\displaystyle \frac{x^3}{3}+2x^2-x+C

\displaystyle \frac{x^3}{3}+2x^2-2x

Correct answer:

\displaystyle \frac{x^3}{3}+2x^2-2x+C

Explanation:

First, chop up the fraction into three simplified terms:
\displaystyle \int x^2+4x-2dx

Now integrate.

\displaystyle \frac{x^3}{3}+\frac{4x^2}{2}-2x=\frac{x^3}{3}+2x^2-2x

Now add a C because it is an indefinite integral:

\displaystyle \frac{x^3}{3}+2x^2-2x+C

Example Question #2542 : Calculus Ii

Calculate the following integral: \displaystyle \int cos^{2}x+x\sqrt{x^{2}+1}dx

Possible Answers:

\displaystyle \frac{1}{2}x+\frac{1}{4}sin(2x)+(\sqrt{x^{2}+1})^{3}+C

\displaystyle \frac{1}{4}sin(2x)+\frac{1}{3}(\sqrt{x^{2}+1})^{3}+C

\displaystyle \frac{1}{2}x+\frac{1}{4}sin(x)+\frac{1}{3}(\sqrt{x^{2}+1})^{3}+C

\displaystyle \frac{1}{2}x+\frac{1}{4}sin(2x)+\frac{1}{3}(\sqrt{x^{2}+1})^{3}+C

Correct answer:

\displaystyle \frac{1}{2}x+\frac{1}{4}sin(2x)+\frac{1}{3}(\sqrt{x^{2}+1})^{3}+C

Explanation:

Separate integral into two separate integrals:

\displaystyle \int cos^{2}x+x\sqrt{x^{2}+1}dx=\int cos^{2}xdx+\int x\sqrt{x^{2}+1}dx  

Solve \displaystyle \int cos^{2}xdx first. 

Use power-reducing identity to simplify integral: \displaystyle \int \frac{1+cos(2x)}{2}dx.

Factor  \displaystyle \frac{1}{2} out of the integral: \displaystyle \int \frac{1}{2}+\frac{cosx}{2}dx

Separate into two integrals: \displaystyle \frac{1}{2}\int 1dx+\frac{1}{2}\int cos (2x)dx=\frac{1}{2}x+\frac{1}{2}\int cos (2x)dx.

Use the following substitution for the second integral: \displaystyle u=2x \displaystyle du=2 \displaystyle \frac{1}{2}du=dx

Plug in substitution and solve:  \displaystyle \frac{1}{4}\int cosudu=\frac{1}{4}sinu+C=\frac{1}{4}sin(2x)+C  \displaystyle \frac{1}{4}sinu=\frac{1}{4}sin(2x)+C.

Therefore: \displaystyle \int cos^{2}x=\frac{1}{2}x+\frac{1}{4}sin(2x)+C

Solve \displaystyle \int x\sqrt{x^{2}+1}dx

Make the following substitution: \displaystyle u=x^{2}+1  \displaystyle du=2xdx   \displaystyle \frac{1}{2}du=dx. Plug in substitution and solve: \displaystyle \frac{1}{2}\int \sqrt{u}du=(\frac{2}{3})\frac{1}{2}(x^{2}+1)^{\frac{3}{2}}+C=\frac{1}{3}(x^{2}+1)^{\frac{3}{2}}+C

Combine answers to two original integrals: \displaystyle \int cos^{2}xdx+\int x\sqrt{x^{2}+1}dx=\frac{1}{2}x+\frac{1}{4}sin(2x)+\frac{1}{3}(x^{2}+1)^{\frac{3}{2}}+C

\displaystyle \int cos^{2}x+x\sqrt{x^{2}+1}dx=\frac{1}{2}x+\frac{1}{4}sin(2x)+\frac{1}{3}(x^{2}+1)^{\frac{3}{2}}+C

Example Question #791 : Integrals

Evaluate.

\displaystyle \int 44x^{43} \ dx

Possible Answers:

\displaystyle F(x) = 43x^{44 }+ C

\displaystyle F(x) = x^{43}+ C

\displaystyle F(x) = x^{44 }+ C

Answer not listed

\displaystyle F(x) = \frac{1}{44}x^{44 }+ C

Correct answer:

\displaystyle F(x) = x^{44 }+ C

Explanation:

\displaystyle \int f(x) dx

In order to evaluate this integral, first find the antiderivative of \displaystyle f(x).

If \displaystyle f(x) = a then \displaystyle F(x) = ax + C

If \displaystyle f(x) = x a then \displaystyle F(x) = \frac{x^{a+1}}{a+1} + C

If \displaystyle f(x) = \frac{1}{x} then \displaystyle F(x) = ln(x) + C

If \displaystyle f(x) = e ^x then \displaystyle F(x) = e^x + C

If \displaystyle f(x) = \cos(x) then \displaystyle F(x) = \sin(x) + C

If \displaystyle f(x) = \sin(x) then \displaystyle F(x) = - \cos(x) + C

If \displaystyle f(x) = \sec^2 (x) then \displaystyle F(x) = \tan(x) + C

 

In this case, \displaystyle f(x) = 44x^{43}.

The antiderivative is  \displaystyle F(x) = x^{44 }+ C.

Example Question #792 : Integrals

Evaluate.

\displaystyle \int \frac{1}{4x-19} \ dx

Possible Answers:

\displaystyle F(x) = \frac{4x-19}{4} + C

\displaystyle F(x) = 4x-19 + C

\displaystyle F(x) = \frac{4}{4x-19} + C

\displaystyle F(x) = \frac{x-4}{4x-19} + C

Answer not listed.

Correct answer:

\displaystyle F(x) = \frac{4x-19}{4} + C

Explanation:

\displaystyle \int f(x) dx

In order to evaluate this integral, first find the antiderivative of \displaystyle f(x).

If \displaystyle f(x) = a then \displaystyle F(x) = ax + C

If \displaystyle f(x) = x a then \displaystyle F(x) = \frac{x^{a+1}}{a+1} + C

If \displaystyle f(x) = \frac{1}{x} then \displaystyle F(x) = ln(x) + C

If \displaystyle f(x) = e ^x then \displaystyle F(x) = e^x + C

If \displaystyle f(x) = \cos(x) then \displaystyle F(x) = \sin(x) + C

If \displaystyle f(x) = \sin(x) then \displaystyle F(x) = - \cos(x) + C

If \displaystyle f(x) = \sec^2 (x) then \displaystyle F(x) = \tan(x) + C

 

In this case, \displaystyle f(x) = \frac{1}{4x-19}.

The antiderivative is  \displaystyle F(x) = \frac{4x-19}{4} + C.

Example Question #2541 : Calculus Ii

Integrate to lowest terms: \displaystyle \large \int \sqrt {x}-1

Possible Answers:

\displaystyle \large \large \frac {x^\frac {3}{2}-\frac {3}{2}x}{\frac {3}{2}}+C

\displaystyle \large \large \frac {2x^\frac {3}{2}+{3}x} {3}+C

\displaystyle \large \large \frac {2x^\frac {3}{2}-{3}x} {3}+C

None of the Above

Correct answer:

\displaystyle \large \large \frac {2x^\frac {3}{2}-{3}x} {3}+C

Explanation:

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Example Question #794 : Integrals

Evaluate.

\displaystyle \int e^{8x-43}\ dx

Possible Answers:

Answer not listed.

\displaystyle F(x) = \frac{8x}{e^{8x-43}} + C

\displaystyle F(x) = \frac{8}{e^{8x-43}} + C

\displaystyle F(x) = \frac{e^{8x-43}}{8} + C

Correct answer:

\displaystyle F(x) = \frac{e^{8x-43}}{8} + C

Explanation:

\displaystyle \int f(x) dx

In order to evaluate this integral, first find the antiderivative of \displaystyle f(x).

If \displaystyle f(x) = a then \displaystyle F(x) = ax + C

If \displaystyle f(x) = x a then \displaystyle F(x) = \frac{x^{a+1}}{a+1} + C

If \displaystyle f(x) = \frac{1}{x} then \displaystyle F(x) = ln(x) + C

If \displaystyle f(x) = e ^x then \displaystyle F(x) = e^x + C

If \displaystyle f(x) = \cos(x) then \displaystyle F(x) = \sin(x) + C

If \displaystyle f(x) = \sin(x) then \displaystyle F(x) = - \cos(x) + C

If \displaystyle f(x) = \sec^2 (x) then \displaystyle F(x) = \tan(x) + C

 

In this case, \displaystyle f(x)= e^{8x-43}.

The antiderivative is  \displaystyle F(x) = \frac{e^{8x-43}}{8} + C.

Example Question #795 : Integrals

Evaluate.

\displaystyle \int \cos (12) \ dx

Possible Answers:

\displaystyle F(x) = \frac{\cos(12)}{x} + C

\displaystyle F(x) = \cos(12x) + C

\displaystyle F(x) = \cos(12)x + C

\displaystyle F(x) = \cos(12) + C

Answer not listed.

Correct answer:

\displaystyle F(x) = \cos(12)x + C

Explanation:

\displaystyle \int f(x) dx

In order to evaluate this integral, first find the antiderivative of \displaystyle f(x).

If \displaystyle f(x) = a then \displaystyle F(x) = ax + C

If \displaystyle f(x) = x a then \displaystyle F(x) = \frac{x^{a+1}}{a+1} + C

If \displaystyle f(x) = \frac{1}{x} then \displaystyle F(x) = ln(x) + C

If \displaystyle f(x) = e ^x then \displaystyle F(x) = e^x + C

If \displaystyle f(x) = \cos(x) then \displaystyle F(x) = \sin(x) + C

If \displaystyle f(x) = \sin(x) then \displaystyle F(x) = - \cos(x) + C

If \displaystyle f(x) = \sec^2 (x) then \displaystyle F(x) = \tan(x) + C

 

In this case, \displaystyle f(x)= \cos(12).

The antiderivative is  \displaystyle F(x) = \cos(12)x + C.

Example Question #171 : Indefinite Integrals

Evaluate.

\displaystyle \int \sin(10x-3) \ dx

Possible Answers:

\displaystyle F(x) = -\frac{\cos(10x-3)}{10} + C

\displaystyle F(x) = -\frac{\cos(10x)}{\sin(10x-3)} + C

\displaystyle F(x) = -10\cos(10x-3) + C

Answer not listed.

\displaystyle F(x) = -\frac{10}{\sin(10x-3)} + C

Correct answer:

\displaystyle F(x) = -\frac{\cos(10x-3)}{10} + C

Explanation:

\displaystyle \int f(x) dx

In order to evaluate this integral, first find the antiderivative of \displaystyle f(x).

If \displaystyle f(x) = a then \displaystyle F(x) = ax + C

If \displaystyle f(x) = x a then \displaystyle F(x) = \frac{x^{a+1}}{a+1} + C

If \displaystyle f(x) = \frac{1}{x} then \displaystyle F(x) = ln(x) + C

If \displaystyle f(x) = e ^x then \displaystyle F(x) = e^x + C

If \displaystyle f(x) = \cos(x) then \displaystyle F(x) = \sin(x) + C

If \displaystyle f(x) = \sin(x) then \displaystyle F(x) = - \cos(x) + C

If \displaystyle f(x) = \sec^2 (x) then \displaystyle F(x) = \tan(x) + C

 

In this case, \displaystyle f(x)= \sin(10x-3).

The antiderivative is  \displaystyle F(x) = -\frac{\cos(10x-3)}{10} + C.

Example Question #172 : Indefinite Integrals

Evaluate.

\displaystyle \int_{1}^{2}\cos (12) \ dx

Possible Answers:

Answer not listed.

\displaystyle 5.32

\displaystyle 0.98

\displaystyle -0.53

\displaystyle 2.34

Correct answer:

\displaystyle 0.98

Explanation:

\displaystyle \int_{a}^{b}f(x) \ dx

In order to evaluate this integral, first find the antiderivative of \displaystyle f(x).

In this case, \displaystyle f(x)= \cos(12).

The antiderivative is  \displaystyle F(x) = \cos(12)x.

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative: 

\displaystyle \int_{1}^{2}\cos (12) \ dx = \left ( \cos(12)x \right )_{1}^{2}

\displaystyle = \cos(12)(2) - \left ( \cos(12)(1) \right )

\displaystyle = 0.98

Example Question #451 : Finding Integrals

Evaluate.

\displaystyle \int_{1}^{2} \sin(10x-3) \ dx

Possible Answers:

\displaystyle 5.8

\displaystyle 0.1

\displaystyle 1.3

Answer not listed

\displaystyle 2.9

Correct answer:

Answer not listed

Explanation:

\displaystyle \int_{a}^{b}f(x) \ dx

In order to evaluate this integral, first find the antiderivative of \displaystyle f(x).

In this case, \displaystyle f(x)= \sin(10x-3).

The antiderivative is  \displaystyle F(x) = -\frac{\cos(10x-3)}{10}.

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative: 

\displaystyle \int_{1}^{2} \sin(10x-3) \ dx = \left ( -\frac{\cos(10x-3)}{10} \right )_{1}^{2}

\displaystyle = -\frac{\cos(10(2)-3)}{10} - \left ( -\frac{\cos(10-3)}{10} \right )

\displaystyle = 0.004

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