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Calculus 2 : Indefinite Integrals

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Example Questions

Example Question #161 : Indefinite Integrals

Solve the indefinite integral.

$\int sec^2x\,dx$

Possible Answers:

$sin\,x+c$

$cos\,x+c$

$cot\,x+c$

$tan\,x+c$

Correct answer:

$tan\,x+c$

Explanation:

The antiderivative of sec^2x is $tan\,x$ .

As such,
$\int sec^2x\,dx=tan\,x+c$

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Example Question #2539 : Calculus Ii

$\int x^3+2x$

Possible Answers:

$\frac {x^4}{4}+x^2+C$

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

None of the Above

$\frac {x^4}{4}+x^3+C$

Correct answer:

$\frac {x^4}{4}+x^2+C$

Explanation:

Step 1: We will first take the antiderivative of x^3 .. To do this, we add to the exponent. We then divide the entire term by the new exponent..

So, $x^3=\frac {x^{3+1}}{3+1}=\frac {x^4}{4}$

Step 2: We will take the antiderivative of 2x. We will apply the same rules that we used in Step 1...

$2x=2 \cdot \frac {x^{1+1}}{1+1}=2 \cdot \frac {x^2}{2}=x^2$

We will add the antiderivatives from both steps together..

(Recall that when taking the indefinite integral of a function, +C must be added to the integrated function as it represents any constants that may be present.)

The final answer is:

$\frac {x^4}{4}+x^2+C$

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Example Question #2531 : Calculus Ii

$\int \frac{3x+1}{x}dx$

Possible Answers:

$3x+ln\left | x \right |$

$ln\left | x \right |+C$

$3x+4ln\left | x \right |+C$

$3x+ln\left | 3x \right |+C$

$3x+ln\left | x \right |+C$

Correct answer:

$3x+ln\left | x \right |+C$

Explanation:

First, chop up the fraction into two separate terms:
$\int 3+\frac{1}{x}dx$

Now integrate:

$3x+ln\left | x \right |$

Add a C because it is an indefinite integral:

$3x+ln\left | x \right |+C$

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Example Question #442 : Finding Integrals

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

Possible Answers:

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

x^3+2x^2-2x+C

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

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

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

Correct answer:

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

Explanation:

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

Now integrate.

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

Now add a C because it is an indefinite integral:

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

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Example Question #2542 : Calculus Ii

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Separate integral into two separate integrals:

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

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

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

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

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

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

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

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

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

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

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

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

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Example Question #791 : Integrals

Evaluate.

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

Possible Answers:

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

$F(x) = x^{43}+ C$

$F(x) = x^{44 }+ C$

Answer not listed

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

Correct answer:

$F(x) = x^{44 }+ C$

Explanation:

$\int f(x) dx$

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

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

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

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

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

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

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

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

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

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

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Example Question #792 : Integrals

Evaluate.

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

Possible Answers:

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

F(x) = 4x-19 + C

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

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

Answer not listed.

Correct answer:

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

Explanation:

$\int f(x) dx$

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

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

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

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

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

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

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

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

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

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

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Example Question #441 : Finding Integrals

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

Possible Answers:

None of the Above

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

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

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

Correct answer:

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

Explanation:

Screen shot 2016 01 02 at 10.25.11 am

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Example Question #794 : Integrals

Evaluate.

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

Possible Answers:

Answer not listed.

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

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

$F(x) = 8e^{8x-43}} + C$

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

Correct answer:

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

Explanation:

$\int f(x) dx$

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

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

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

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

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

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

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

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

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

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

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Example Question #795 : Integrals

Evaluate.

$\int \cos (12) \ dx$

Possible Answers:

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

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

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

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

Answer not listed.

Correct answer:

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

Explanation:

$\int f(x) dx$

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

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

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

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

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

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

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

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

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

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

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Calculus 2 : Indefinite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #161 : Indefinite Integrals

Example Question #2539 : Calculus Ii

Example Question #2531 : Calculus Ii

Example Question #442 : Finding Integrals

Example Question #2542 : Calculus Ii

Example Question #791 : Integrals

Example Question #792 : Integrals

Example Question #441 : Finding Integrals

Example Question #794 : Integrals

Example Question #795 : Integrals

All Calculus 2 Resources

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