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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #342 : Finding Integrals

Evaluate.

$\int -\cos(7x) \ dx$

Possible Answers:

$F(x) = - \sin(7x)$

Answer not listed.

$F(x) = \frac{\cos(7x)}{7}$

$F(x) = - \frac{\sin(7x)}{7}$

$F(x) = \frac{\sin(x)}{7}$

Correct answer:

$F(x) = - \frac{\sin(7x)}{7}$

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, $f(x) =-\cos(7x)$ .

The antiderivative is $F(x) = - \frac{\sin(7x)}{7}$ .

Report an Error

Example Question #343 : Finding Integrals

Evaluate.

$\int e^{2x-5} - \sin(2x-5) \ dx$

Possible Answers:

$F(x) = \frac{e^{2x-5}}{e^2} + \frac{\cos(2x-5)}{\cos2}$

Answer not listed.

$F(x) = e^{2x-5}+ \cos(2x-5)$

$F(x) = \frac{e^{2x-5}}{x} - \frac{\cos(2x-5)}{x}$

$F(x) = \frac{e^{2x-5}}{2} + \frac{\cos(2x-5)}{2}$

Correct answer:

$F(x) = \frac{e^{2x-5}}{2} + \frac{\cos(2x-5)}{2}$

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, $f(x) =e^{2x-5} - \sin(2x-5)$ .

The antiderivative is $F(x) = \frac{e^{2x-5}}{2} + \frac{\cos(2x-5)}{2}$ .

Report an Error

Example Question #344 : Finding Integrals

Evaluate.

$\int 5x^4 + 12x \ dx$

Possible Answers:

F(x) = x^5 + 6x^2

$F(x) = \frac{1}{5}x^5 + \frac{1}{12}x^2$

F(x) = 5x^5 + 12x^2

Answer not listed.

F(x) = 5x^4 + 12x

Correct answer:

F(x) = x^5 + 6x^2

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) =5x^4 + 12x .

The antiderivative is F(x) = x^5 + 6x^2 .

Report an Error

Example Question #345 : Finding Integrals

Evaluate.

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

Possible Answers:

$F(x) = \frac{1}{ \ln(x-1)}$

$F(x) = 4 \ln(x-1)$

$F(x) =\frac{\ln(x-1)}{4}$

$F(x) = \ln(x-1)$

Answer not listed.

Correct answer:

$F(x) = 4 \ln(x-1)$

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, $f(x) = \frac{4}{x-1}$ .

The antiderivative is $F(x) = 4 \ln(x-1)$ .

Report an Error

Example Question #346 : Finding Integrals

Evaluate.

$\int 12 \ dx$

Possible Answers:

F(x) = 12x

F(x) = 12x^2

$F(x) = \frac{1}{12}x$

F(x) =0

Answer not listed

Correct answer:

F(x) = 12x

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) = 12 .

The antiderivative is F(x) = 12x .

Report an Error

Example Question #347 : Finding Integrals

Evaluate.

$\int -\sin(6x) +3 \ dx$

Possible Answers:

$F(x) = \frac{\cos(x)}{6x} + 3$

$F(x) = \frac{\cos(6x)}{6} + 3x$

$F(x) = -\cos(6x) + 3x$

Answer not listed.

$F(x) =- \frac{\cos(6x)}{6} + 3x$

Correct answer:

$F(x) = \frac{\cos(6x)}{6} + 3x$

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, $f(x) = -\sin(6x) +3$ .

The antiderivative is $F(x) = \frac{\cos(6x)}{6} + 3x$ .

Report an Error

Example Question #348 : Finding Integrals

Evaluate.

$\int 2x - \frac{1}{3x} \ dx$

Possible Answers:

$F(x) = 2x^2 - \frac{\ln(3x)}{3}$

$F(x) = \frac{1}{2}x^2 - \ln(x)$

Answer not listed.

$F(x) = x^2 - \frac{\ln(x)}{3}$

$F(x) = 2x^2 - \ln(x)$

Correct answer:

$F(x) = x^2 - \frac{\ln(x)}{3}$

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, $f(x) = 2x - \frac{1}{3x}$ .

The antiderivative is $F(x) = x^2 - \frac{\ln(x)}{3}$ .

Report an Error

Example Question #71 : Indefinite Integrals

Evaluate.

$\int e^{7x} - 7 \ dx$

Possible Answers:

Answer not listed.

$F(x) = \frac{e^{7x}}{7} - 7x$

$F(x) = 7e^{7x} - 7$

$F(x) = 7e^{7x}$

$F(x) = e^{7x} - 7$

Correct answer:

$F(x) = \frac{e^{7x}}{7} - 7x$

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, $f(x) = e^{7x} + 7$ .

The antiderivative is $F(x) = \frac{e^{7x}}{7} - 7x$ .

Report an Error

Example Question #72 : Indefinite Integrals

Evaluate.

$\int \sin(x) + 6 - \frac{1}{5x}\ dx$

Possible Answers:

$F(x) = - \cos(x) + 6x - \frac{\ln (x)}{5}$

$F(x) = - \sin(x) + 6x - \frac{\ln (5x)}{5}$

Answer not listed.

$F(x) = - \cos(x) + 6 - \frac{\ln (5x)}{5}$

$F(x) = - \cos(x) + 6x - 5\ln (x)$

Correct answer:

$F(x) = - \cos(x) + 6x - \frac{\ln (x)}{5}$

Explanation:

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, $f(x) = \sin(x) + 6 - \frac{1}{5x}$ .

The antiderivative is $F(x) = - \cos(x) + 6x - \frac{\ln (x)}{5}$ .

Report an Error

Example Question #2441 : Calculus Ii

$\int \frac{2x^2-8x+2}{2}dx$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

First, make the fraction three separate terms: $\int x^2-4x+1dx$ . Now, integrate as normal, remembering to raise the exponent by 1 and then also putting that result on the denominator: $\frac{x^3}{3}-2x^2+x$ . Remember to add a C at the end because it is an indefinite integral: $\frac{x^3}{3}-2x^2+x+C$ .

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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #342 : Finding Integrals

Example Question #343 : Finding Integrals

Example Question #344 : Finding Integrals

Example Question #345 : Finding Integrals

Example Question #346 : Finding Integrals

Example Question #347 : Finding Integrals

Example Question #348 : Finding Integrals

Example Question #71 : Indefinite Integrals

Example Question #72 : Indefinite Integrals

Example Question #2441 : Calculus Ii

All Calculus 2 Resources

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