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Calculus 2 : Indefinite Integrals

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Example Question #251 : Indefinite Integrals

Integrate the following function:

$\int (x^3-3x^2+4x)dx$ .

Possible Answers:

3x^2-6x+2

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

3x^2-6x+2+C

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

Correct answer:

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

Explanation:

For integration, we do the opposite of a derivative. Using the power rule, we increase the exponent by one and divide by the new exponent.

$\int (x^3-3x^2+4x)dx=$

$\frac{x^{3+1}}{3+1}-3\frac{x^{2+1}}{2+1}+4\frac{x^{1+1}}{1+1}+C$ .

Simplifying, we get:

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

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

Remember, because this is an indefinite integral, we have to add the constant at the end ().

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Example Question #521 : Finding Integrals

Calculate the value of the following integral:

$\int_{0}^{+\infty}e^{-x}dx$

Possible Answers:

1/e

$\infty$

Correct answer:

Explanation:

$\int_{0}^{+\infty}e^{-x}dx=\lim_{t\to\infty}\int_{0}^{t}e^{-x}dx=\lim_{t\to\infty}\left [ e^0-e^{-t}\right ]= 1$

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Example Question #253 : Indefinite Integrals

Evaluate:

$\int x^{4}e^{x^{5}}dx$

Possible Answers:

$4x^{3}+5e^{x^{4}}+c$

$e^{x^{5}}+c$

$4x^{}{5}e^{x}+c$

$\frac{1}{5} e^{x^{5}}+c$

$20x^{3}e^{x^{4}}+c$

Correct answer:

$\frac{1}{5} e^{x^{5}}+c$

Explanation:

For this problem we have to use substitution to solve the integral. set u= $x^{5}$ , then du= $5x^{4}dx$ . then we see that $\frac{1}{5}du=x^{4}dx$ .

Therefore:

$\int x^{4}e^{x^{5}}dx =$ $\frac{1}{5}\int e^{u}du=$ $\frac{1}{5}e^{u}+c = \frac{1}{5}e^{x^{5}} +c$

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Example Question #881 : Integrals

$\int2x^2+x-3 dx$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

To find this integral, look at each term separately.

For the first term, raise the exponent by 1 and also put that result on the denominator: $2(\frac{x^3}{3})=\frac{2x^3}{3}$ .

For the next term, do the same: $\frac{x^2}{2}$ .

Same for the third term (since it's a constant, multiply the coefficient by x): -3x .

Put those all together to get: $\frac{2x^3}{3}+\frac{x^2}{2}-3x$ . Since this is an indefinite integral, make sure to add C at the end: $\frac{2x^3}{3}+\frac{x^2}{2}-3x+C$ .

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Example Question #882 : Integrals

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

First chop up the fraction into two separate terms and simplify: $\frac{3x^2-4x}{x}=\frac{3x^2}{x}-\frac{4x}{x}=3x-4$ .

Now, integrate that expression. Remember to raise the exponent by 1 and also put that result on the denominator:

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

Since it's an indefinite integral, remember to add C at the end: $\frac{3x^2}{2}-4x+C$ .

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Calculus 2 : Indefinite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #251 : Indefinite Integrals

Example Question #521 : Finding Integrals

Example Question #253 : Indefinite Integrals

Example Question #881 : Integrals

Example Question #882 : Integrals

All Calculus 2 Resources

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