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Calculus 1 : Equations

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Example Questions

Example Question #181 : Equations

Solve the following integral, where a and b are constants:

$\int abe^{b^2x}dx$

Possible Answers:

$\frac{ae^{b^2x}}{b}+C$

$ae^{b^2x}+C$

$\frac{2ae^{b^2x}}{b}+C$

$\frac{1}{2}ae^{b^2x}+C$

Correct answer:

$\frac{ae^{b^2x}}{b}+C$

Explanation:

Keeping in mind that a and b are only constants, the integral is equal to

$\int abe^{b^2x}dx=\frac{abe^{b^2x}}{b^2}+C=\frac{ae^{b^2x}}{b}+C$

and was found using the following rule:

$\int e^{ax}dx=\frac{e^{ax}}{a}+C$

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Example Question #182 : Integral Expressions

Evaluate the following integral:

$\int xy\cos(x^2y)dx$

Possible Answers:

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

$-\frac{1}{2}\sin(x^2y)+C$

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

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

Correct answer:

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

Explanation:

To evaluate the integral, we must make the following substitution:

u=x^2y , du=2xydx

The derivative was found using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Rewrite the integral in terms of u and integrate:

$\frac{1}{2}\int \cos(u)du=\frac{1}{2}\sin(u)+C$

The integral was found using the following rule:

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

Finally, replace u with our original term:

$\frac{1}{2}\sin(x^2y)+C$ .

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Example Question #2262 : Calculus

In circuits with a resistor, the equation for voltage drop is given by:

$\frac{d}{dx}V =\frac{d}{dt}Q*R$ , where is voltage, is charge, and is resistance.

Write the equation as an integral expression for .

Possible Answers:

$Q=\int {\frac{d}{dx}V}R dt$

$Q=\int VR dt$

$Q=\int \frac{V}{R} dt$

$Q=\int \frac{\frac{d}{dx}V}{R} dt$

Correct answer:

$Q=\int \frac{\frac{d}{dx}V}{R} dt$

Explanation:

Although this may seem really difficult, we only need to solve for .

$\frac{d}{dx}V =\frac{d}{dt}Q*R$

$\frac{d}{dt}Q=\frac{\frac{\mathrm{d} }{\mathrm{d} x}V}{R}$

To solve for , integrate both sides:

$Q=\int \frac{\frac{d}{dx}V}{R} dt$

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Example Question #182 : Equations

$\int_{0}^{2}2x^2-5x+1 dx$

Possible Answers:

$-\frac{8}{3}$

$\frac{8}{3}$

$-\frac{5}{3}$

$-\frac{11}{3}$

Correct answer:

$-\frac{8}{3}$

Explanation:

When integrating, remember to add one to the exponent and then put that result on the denominator: $\frac{2x^3}{3}-\frac{5x^2}{2}+x$ . Now evaluate at 2, and then 0. Then subtract the two results. $(\frac{16}{3}-10+2)-0=-\frac{8}{3}$ .

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Example Question #184 : Integral Expressions

$\int \frac{24x^3-2x^2+4x}{4x}dx$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step here is to chop this up into three separate terms and then simplify since we have only one denominator: $\int 6x^2-\frac{1}{2}x+1dx$ . Then, integrate each term, remembering to add one to the exponent and then put that result on the denominator: $6(\frac{x^3}{3})-\frac{1}{2}(\frac{x^2}{2})+x$ . Simplify to get your answer: $2x^3-\frac{x^2}{4}+x+C$ . Remember to add C because it is an indefinite integral.

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Example Question #183 : How To Find Integral Expressions

$\int \frac{9+x}{x}dx$

Possible Answers:

$9ln\left | x \right |+x+C$

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

$\frac{1}{9}ln\left | x \right |+x+C$

$9ln\left | x \right |-x+C$

Correct answer:

$9ln\left | x \right |+x+C$

Explanation:

First, chop this expression up into two terms: $\int \frac{9}{x}+1dx$ . Then, integrate each term, remembering that when there is a single x on a denominator, the integral is $ln\left | x \right |$ . Therefore, the integration is: $9ln\left | x \right |+x+C$ . Remember to add C because it is an indefinite integral.

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Example Question #183 : Equations

$\int_{0}^{3}2x^3-3x^2+xdx$

Possible Answers:

108

Correct answer:

Explanation:

First, integrate each term separately. Remember, when integrating, raise the exponent by one and then also put that result on the denominator: $2(\frac{x^4}{4})-3(\frac{x^3}{3})+\frac{x^2}{2}$ . Then evaluate at 3 and then 0. Subtract two results to get: $(\frac{81}{2}-27+\frac{9}{2})-0=18$ .

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Example Question #185 : How To Find Integral Expressions

$\int \sqrt{x}dx$

Possible Answers:

$\frac{2}{3}$

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

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

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

Correct answer:

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

Explanation:

To integrate this expression, I would first rewrite it to be $\int x^{\frac{1}{2}}dx$ . Then, add one to the exponent and then put that result on the denominator: $\frac{x^{\frac{3}{2}}}{\frac{3}{2}}$ . Then, simplify to get your answer of $\frac{2}{3}x^{\frac{3}{2}}+C$ . Remember to add C because it is an indefinite integral.

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Example Question #1 : How To Write Equations

Given the one-to-one equation f(x)=3x+1, the inverse function f^-1(y)=

Possible Answers:

3x-1

undefined

(y-1)/3

(x-1)/3

(y+1)/3

Correct answer:

(y-1)/3

Explanation:

Before we solve the problem by computation, let's look at the answer choices and see if we can eliminate any answer choices. We know that an inverse function must exist because f(x) is one-to-one, so we can eliminate the answer choice "undefined." Next, we know that the inverse function has to be in terms of y, so we can eliminate the two answer choices with an "x."

Now we can look at the two remaining answer choices. Let y=f(x) and solve for x to find the inverse.

So f(x)=y=3x+1. Solve for x.

x=(y-1)/3

Therefore our answer is (y-1)/3.

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Example Question #1 : Other Writing Equations

Describe the graph of the polar equation r=-6.

Possible Answers:

line with slope 6

line with slope -6

circle with radius 36

circle with diameter 36

circle with radius 6

Correct answer: circle with radius 6

Explanation:

The polar equation formula here is x²+y²=r².

This is the formula for a cirlce, so we can eliminate the two "line" answers.

Plugging in r=-6 into the equation gives x²+y²=36, which describes the graph of a circle with radius 6.

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Calculus 1 : Equations

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #181 : Equations

Example Question #182 : Integral Expressions

Example Question #2262 : Calculus

Example Question #182 : Equations

Example Question #184 : Integral Expressions

Example Question #183 : How To Find Integral Expressions

Example Question #183 : Equations

Example Question #185 : How To Find Integral Expressions

Example Question #1 : How To Write Equations

Example Question #1 : Other Writing Equations

All Calculus 1 Resources

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