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Calculus 1 : Equations

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Example Questions

Example Question #2111 : Calculus

Evaluate $\int_{0}^{3} 3x^2+6dx$ .

Possible Answers:

Correct answer:

Explanation:

Using the Power Rule ( $\int u^ndu={\frac{1}{n+1}}u^{n+1}$ ), take the anti-derivative:

$\int_{0}^{3}3x^2+6dx=3(\frac{1}{2+1}x^{2+1})+6x|_{0}^{3}$

$\int_{0}^{3}3x^2+6dx=3(\frac{1}{3}x^3)+6x|_{0}^{3}$

$\int_{0}^{3}3x^2+6dx=x^3+6x|_{0}^{3}$

Next, evaluate the integral at the given points:

$x^3+6x|_{0}^{3}=[(3)^3+6(3)]-[(0)^3+6(0)]$

$x^3+6x|_{0}^{3}=[27+18]-[0]$

$x^3+6x|_{0}^{3}=45-0$

$x^3+6x|_{0}^{3}=45$

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Example Question #2111 : Calculus

Evaluate $\int_{-3}^{3}2x^2-1dx$ .

Possible Answers:

Correct answer:

Explanation:

Using the Power Rule ( $\int u^ndu={\frac{1}{n+1}}u^{n+1}$ ), take the antiderivative:

$\int_{-3}^{3}2x^2-1dx=[2(\frac{1}{2+1}x^{2+1})-1(x)]_{-3}^{3}$

$\int_{-3}^{3}2x^2-1dx=[2(\frac{1}{3}x^3)-x]_{-3}^3$

$\int_{-3}^{3}2x^2-1dx=[\frac{2}{3}x^3-x]_{-3}^3$

Next, evaluate the integral at the given points:

$[\frac{2}{3}x^3-x]_{-3}^3= [\frac{2}{3}(3)^3-(3)]-[\frac{2}{3}(-3)^3-(-3)]$

$[\frac{2}{3}x^3-x]_{-3}^3=15-(-15)$

$[\frac{2}{3}x^3-x]_{-3}^3=30$

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Example Question #31 : Equations

Evaluate $\int_{-1}^{3}4x^3+2xdx$ .

Possible Answers:

Correct answer:

Explanation:

Using the Power Rule ( $\int u^ndu={\frac{1}{n+1}}u^{n+1}$ ), take the anti-derivative then plug in for the interval given:

$\int_{-1}^{3}4x^3+2xdx=[4[\frac{1}{1+3}x^{1+3}]+2[\frac{1}{1+1}x^{1+1}]]_{-1}^{3}$

$\int_{-1}^{3}4x^3+2xdx=[4(\frac{1}{4}x^4)+2(\frac{1}{2}x^2)]_{-1}^{3}$

$\int_{-1}^{3}4x^3+2xdx=[x^4+x^2]_{-1}^{3}$

$\int_{-1}^{3}4x^3+2xdx=[(3)^4+(3)^2]-[(-1)^4+(-1)^2]$

$\int_{-1}^{3}4x^3+2xdx=[81+9]-[1+1]$

$\int_{-1}^{3}4x^3+2xdx=90-2$

$\int_{-1}^{3}4x^3+2xdx=88$

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Example Question #31 : Integral Expressions

Integrate $\int 6x^4+3x^2+13dx$ .

Possible Answers:

6x^5+3x^3+13x

6x^6+x^3+13x

x^5+x^3+13

$\frac{6}{5}x^5+x^3+13x+C$

Correct answer:

$\frac{6}{5}x^5+x^3+13x+C$

Explanation:

Using the Power Rule ( $\int u^ndu={\frac{1}{n+1}}u^{n+1}$ ), take the anti-derivative:

$\int 6x^4+3x^2+13=6(\frac{1}{4+1})(x^{4+1})+3(\frac{1}{2+1})(x^{2+1})+13(x)+C$

$\int 6x^4+3x^2+13=6(\frac{1}{5})(x^5)+3\frac{1}{3}x^3+13x+C$

$\int 6x^4+3x^2+13=\frac{6}{5}x^5+x^3=13x+C$

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Example Question #32 : Integral Expressions

Evaluate $\int 4x^6-x^3+2 dx$ .

Possible Answers:

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

$\frac{x^7}{7}-\frac{x^4}{4}+2x$

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

$\frac{4}{7}x^7-{x^4}+8x+C$

Correct answer:

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

Explanation:

Using the Power Rule ( $\int u^ndu={\frac{1}{n+1}}u^{n+1}$ ), take the anti-derivative:

$\int 4x^6-x^3+2 dx=4[\frac{1}{6+1}x^{6+1}]-[\frac{1}{3+1}x^{3+1}]+2x+C$

$\int 4x^6-x^3+2 dx=\frac{4}{7}x^7-\frac{1}{4}x^4+2x+C$

$\int 4x^6-x^3+2 dx=\frac{4}{7}x^7-\frac{x^4}{4}+2x+C$

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Example Question #2112 : Calculus

Evaluate $\int 3x^5+4x^2dx$ .

Possible Answers:

x^6+4x^3+C

$\frac{x^6}{2}+\frac{4}{3}x^3+C$

x^6+x^3

$2x^6+\frac{3}{4}x^3+C$

Correct answer:

$\frac{x^6}{2}+\frac{4}{3}x^3+C$

Explanation:

Take the anti-derivative (integrate using the Power Rule : $\int u^ndu={\frac{1}{n+1}}u^{n+1}$ ) of the expression:

$\int 3x^5+4x^2=3(\frac{1}{1+5}x^{1+5})+4(\frac{1}{1+2}x^{1+2})+C$

$\int 3x^5+4x^2=3(\frac{1}{6}x^6)+4(\frac{1}{3}x^3)+C$

$\int 3x^5+4x^2=\frac{x^6}{2}+\frac{4}{3}x^3+C$

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Example Question #31 : Integral Expressions

Evaluate $\int15x^4+2x+4dx$ .

Possible Answers:

$\frac{x^5}{3}+x^2+4x$

15x^5+2x^2+4x+C

3x^5+x^2+4x+C

9x^5+x^2+4x+C

Correct answer:

3x^5+x^2+4x+C

Explanation:

Take the anti-derivative (integrate using the Power Rule : $\int u^ndu={\frac{1}{n+1}}u^{n+1}$ ) of the expression:

$\int 15x^4+2x+4dx=15(\frac{1}{1+4}x^{1+4})+2(\frac{1}{1+1}x^{1+1})+4(x)+C$

$\int 15x^4+2x+4dx=15(\frac{1}{5}x^5)+2(\frac{1}{2}x^2)+4x+C$

$\int 15x^4+2x+4dx=3x^5+x^2+4x+C$

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Example Question #31 : How To Find Integral Expressions

If f'(x) is defined as $f'(x)=9x^{2}+7x-12$ , what is f(x) ?

Possible Answers:

f(x)=18x+7+C

None of the above

$f(x)=3x^{3}+\frac{7}{2}x^{2}-12x+C$

$f(x)=3x^{3}+\frac{7}{2}x^{2}-12x$

f(x)=18x+7

Correct answer:

$f(x)=3x^{3}+\frac{7}{2}x^{2}-12x+C$

Explanation:

Given the derivative $f'(x)=9x^{2}+7x-12$ , we can find the function f(x) by indefinitely integrating f'(x) in accordance with the Power Rule for Integrals: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ , where $n\neq-1$ and is the arbitrary constant of integration.

Using this rule, we therefore know that $\int f'(x)dx=\int (9x^{2}+7x-12)dx=3x^{3}+\frac{7}{2}x^{2}-12x+C$ .

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Example Question #1084 : Functions

If f'(x) is defined as $f'(x)=3x^{2}-6x+7$ , what is f(x) ?

Possible Answers:

$f(x)=x^{3}-3x^{2}+7x$

f(x)=6x-6+C

f(x)=6x-6

None of the above

$f(x)=x^{3}-3x^{2}+7x+C$

Correct answer:

$f(x)=x^{3}-3x^{2}+7x+C$

Explanation:

Given the derivative $f'(x)=3x^{2}-6x+7$ , we can find the function f(x) by indefinitely integrating f'(x) in accordance with the Power Rule for Integrals: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ , where $n\neq-1$ and is the arbitrary constant of integration.

Using this rule, we therefore know that $\int f'(x)dx=\int (3x^{2}-6x+7)=x^{3}-3x^{2}+7x+C$ .

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Example Question #33 : Integral Expressions

If f'(x) is defined as $f'(x)=5x^{2}-8x+11$ , what is f(x) ?

Possible Answers:

$f(x)=\frac{5}{3}x^{3}-4x^{2}+11x+C$

$f(x)=\frac{5}{3}x^{3}-4x^{2}+11x$

f(x)=10x-8+C

f(x)=10x-8

None of the above

Correct answer:

$f(x)=\frac{5}{3}x^{3}-4x^{2}+11x+C$

Explanation:

Given the derivative $f'(x)=5x^{2}-8x+11$ , we can find the function f(x) by indefinitely integrating f'(x) in accordance with the Power Rule for Integrals: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ , where $n\neq-1$ and is the arbitrary constant of integration.

Using this rule, we therefore know that $\int f'(x)dx=\int (5x^{2}-8x+11)dx=\frac{5}{3}x^{3}-4x^{2}+11x+C$ .

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Calculus 1 : Equations

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #2111 : Calculus

Example Question #2111 : Calculus

Example Question #31 : Equations

Example Question #31 : Integral Expressions

Example Question #32 : Integral Expressions

Example Question #2112 : Calculus

Example Question #31 : Integral Expressions

Example Question #31 : How To Find Integral Expressions

Example Question #1084 : Functions

Example Question #33 : Integral Expressions

All Calculus 1 Resources

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