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Calculus 1 : Integral Expressions

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Example Questions

Example Question #101 : Writing Equations

Evaluate the following indefinite integral:

$\int \frac{2}{x}dx$

Possible Answers:

ln(x)+C

2ln(x)+C

$\frac{1}{2}ln(x)+C$

2ln(2x)+C

Correct answer:

2ln(x)+C

Explanation:

To solve this problem, we have to remember properties of natural logs. Pull the 2 out in front of the integral and solve. Don't forget to add "+C" at the end!

$\\\int \frac{2}{x}dx=\\ \\ 2\int \frac{1}{x}dx=2*ln(x)\\ \\= 2ln(x)+C$

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Example Question #102 : Integral Expressions

Evaluate the following indefinite integral:

$\int \left(\frac{1}{2}x+5\right)dx$

Possible Answers:

$\frac{1}{4}x^{2}-5x+C$

$\frac{1}{4}x^{2}+5x+C$

$\frac{1}{2}x^{2}+5x+C$

$x^{2}+5x+C$

Correct answer:

$\frac{1}{4}x^{2}+5x+C$

Explanation:

To solve this integral, use the power rule. Applying it to this problem gives us the following for the first term:

$\frac{x^{1+1}}{2*(1+1)}=\frac{x^{2}}{4}$

And the following for the second term:

$\frac{5x^{0+1}}{0+1}=\frac{5x}{1}=5x$

We can combine these terms and add our "C" to get the final answer:

$\frac{1}{4}x^{2}+5x+C$

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Example Question #102 : Writing Equations

Evaluate the following integral:

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

Possible Answers:

$\frac{1}{2}\ln(\sqrt{3x+1})+C$

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

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

$-\frac{3}{2}\ln(\sqrt{3x+1})+C$

Correct answer:

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

Explanation:

To integrate, we must make the following subsitution:

$u=\sqrt{3x+1}, du=\frac{3}{2}(3x+1)^{-\frac{1}{2}}dx$

Rearranging to get du in terms of x we get the following.

$du=\frac{3}{2}(3x+1)^{-\frac{1}{2}}dx\rightarrow \frac{2}{3}du=(3x+1)^{-\frac{1}{2}}$

Now, rewrite the integral in terms of u, and integrate:

$\frac{2}{3}\int {du}=\frac{2}{3}u=\frac{2}{3}\sqrt{3x+1}+C$

which was found using the following rule:

$\int dx=x+C$

Now, replace u with our original term:

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

Notice that the absolute value went away, because the square root is always positive.

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Example Question #103 : Writing Equations

Find the integral of the function f(x)=sin(x)cos(x) .

Possible Answers:

$\frac{cos^2(x)}{2}$

2sin(x)

$\frac{sin^2(x)}{2}$

2cos(x)

sin(2x)

Correct answer:

$\frac{sin^2(x)}{2}$

Explanation:

To find this integral, use the method of integration by parts:

$\int udv=uv-\int vdu$

Let u=sin(x),du=cos(x)dx,dv=cos(x)dx,v=sin(x) such that

$\int sin(x)cos(x)dx=sin(x)sin(x)-\int sin(x)cos(x)dx$

There is now an integral on each side; however, it is the same integral. Move the integral on the right to the left:

$2\int sin(x)cos(x)dx=sin(x)sin(x)$

Now recall that we're asked to find the value of $\int sin(x)cos(x)$ , so to find that, we need only divide each side by two:

$\int sin(x)cos(x)dx=\frac{sin^2(x)}{2}$

Ta-da.

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Example Question #103 : Integral Expressions

Evaluate the indefinite integral:

$\int (x^3+2x+\sqrt{x+2})dx$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Begin by rewriting the equation in terms of exponents:

$\int (x^3+2x^1+(x+2)^\frac{1}{2})dx$

Afterwards, add one to each exponent and divide by the resultant value for each term to do the integral; be sure to add a constant of integration:

For this particular problem we will use the Power Rule on each term.

Power Rule: $x^n \rightarrow nx^{n-1}$ .

Appying this rule we find the following function.

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

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

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Example Question #2182 : Calculus

Evaluate the following integral:

$\int 2x^2\cos(x^3)dx$

Possible Answers:

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

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

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

$\sin(x^3)+C$

Correct answer:

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

Explanation:

To integrate, we must perform the following substitution:

u=x^3, du=3x^2dx

Now, rewrite the integral and integrate:

$\frac{2}{3}\int \cos(u)du = \frac{2}{3}\sin(u)+C$

The integration was performed using the following rule:

$\int \cos(x)dx=\sin(x)+C$

Finally, replace u with the original term we designated at the start:

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

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Example Question #104 : Writing Equations

Evaluate the following integral:

$\int \frac{dx}{16x^2+1}$

Possible Answers:

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

$\frac{1}{4}\tan(4x)+C$

$\arctan(4x)+C$

$\frac{1}{16}\arctan(4x)+C$

Correct answer:

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

Explanation:

To integrate, we must perform the following substiution:

u=4x, du=4dx

Now, rewrite the integral and integrate:

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

The integral was performed using the following rule:

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

Finally, replace u with our original, x term:

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

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Example Question #105 : Writing Equations

Evaluate the integral:

$\int 2\cos(x)\sin(x)dx$

Possible Answers:

$2\cos^2(x)+C$

$-\cos^2(x)+C$

$\sin^2(x)+C$

$2\sec^2(x)+C$

Correct answer:

$-\cos^2(x)+C$

Explanation:

To perform the integral, we must use the following substitution:

$u=\cos(x), du=-\sin(x)dx$

Now, rewrite the integral and integrate:

$-2\int udu=-2\frac{u^2}{2}+C$

The integration was performed using the following rule:

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

Now, replace u with the original, x containing term:

$-\cos^2(x)+C$

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Example Question #106 : Integral Expressions

Evaluate the integral:

$\int \sec^2(x)e^{\tan(x)}dx$

Possible Answers:

$e^{\cos(x)}+C$

$e^{\tan(x)}+C$

$e^{\sec(x)}+C$

$\sec(x)+C$

Correct answer:

$e^{\tan(x)}+C$

Explanation:

To integrate, we must perform the following subsitution:

$u=\tan(x), du=\sec^2(x)dx$

Now, rewrite the integral and integrate:

$\int e^udu=e^u+C$

We used the following rule to integrate:

$\int e^x dx=e^x+C$

Finally, replace u with our original term:

$e^{\tan(x)}+C$

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Example Question #2183 : Calculus

Evaluate the integral:

$\int 10x\cos(x^2)dx$

Possible Answers:

$-5\sin^2(x)+C$

$5\cos^2(x)+C$

$10\cos(x^2)+C$

$5\sin(x^2)+C$

Correct answer:

$5\sin(x^2)+C$

Explanation:

To integrate, we must first make the following subsitution:

u=x^2, du=2xdx

Now, rewrite the integral and integrate:

$5\int \cos(u)du=5\sin(u)+C$

We used the following rule to integrate:

$\int \cos(x)dx=\sin(x)+C$

Finally, replace u with the original term:

$5\sin(x^2)+C$

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Calculus 1 : Integral Expressions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #101 : Writing Equations

Example Question #102 : Integral Expressions

Example Question #102 : Writing Equations

Example Question #103 : Writing Equations

Example Question #103 : Integral Expressions

Example Question #2182 : Calculus

Example Question #104 : Writing Equations

Example Question #105 : Writing Equations

Example Question #106 : Integral Expressions

Example Question #2183 : Calculus

All Calculus 1 Resources

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