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Calculus 2 : Indefinite Integrals

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Example Questions

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Example Question #51 : Indefinite Integrals

Evaluate.

$\int e^{2x-4} \ dx$

Possible Answers:

Answer not listed

$F(x) = e^{2x-4}$

$F(x) = \frac{e^{2x-4}}{e^x}$

$F(x) = \frac{e^{2x-4}}{2}$

$F(x) = \frac{e^{2x-4}}{2x-4}$

Correct answer:

$F(x) = \frac{e^{2x-4}}{2}$

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, $f(x) = e^{2x-4}$ and u= 2x-4 .

The antiderivative is $F(x) = \frac{e^{2x-4}}{2}$ .

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Example Question #52 : Indefinite Integrals

Evaluate.

$\int (x-1)^2\ dx$

Possible Answers:

Answer not listed

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

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

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

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

Correct answer:

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

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) = (x-1)^2 .

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

Report an Error

Example Question #53 : Indefinite Integrals

Evaluate.

$\int 6x^2 + \sin(x) \ dx$

Possible Answers:

Answer not listed

$F(x) = 2x^3 + \cos(x)$

$F(x) = x^4 - \cos(x)$

$F(x) = x^4 - \sin(x)$

$F(x) = 2x^3 - \cos(x)$

Correct answer:

$F(x) = 2x^3 - \cos(x)$

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, $f(x) = 6x^2 + \sin(x)$ .

The antiderivative is $F(x) = 2x^3 - \cos(x)$ .

Report an Error

Example Question #54 : Indefinite Integrals

Evaluate.

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

Possible Answers:

F(x) = x^2 - x^3

$F(x) = \frac{1}{4}x - \frac{1}{9} x^2$

$F(x) = \frac{1}{2}x^2 - \frac{1}{3}x^3$

Answer not listed.

$F(x) = \frac{1}{4}x^2 - \frac{1}{9} x^3$

Correct answer:

$F(x) = \frac{1}{4}x^2 - \frac{1}{9} x^3$

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

The antiderivative is $F(x) = \frac{1}{4}x^2 - \frac{1}{9} x^3$ .

Report an Error

Example Question #55 : Indefinite Integrals

Evaluate.

$\int \cos (2x-1) \ dx$

Possible Answers:

$F(x) = 2 \cos (2x-1)$

$F(x) = 2 \sin (2x-1)$

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

Answer not listed.

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

Correct answer:

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

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, $f(x) = \cos (2x-1)$ .

The antiderivative is $F(x) = \frac{\sin (2x-1)}{2}$ .

Report an Error

Example Question #56 : Indefinite Integrals

Evaluate.

$\int \frac{1}{4x-5} \ dx$

Possible Answers:

$F(x) = \frac{ \ln (x)}{4x-5}$

$F(x) = 4\ln (4x-5)$

Answer not listed.

$F(x) = \frac{ \ln (4x-5)}{4}$

$F(x) = \frac{ \ln (4x-5)}{\ln(x)}$

Correct answer:

$F(x) = \frac{ \ln (4x-5)}{4}$

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, $f(x) = \frac{1}{4x-5}$ .

The antiderivative is $F(x) = \frac{ \ln (4x-5)}{4}$ .

Report an Error

Example Question #57 : Indefinite Integrals

Evaluate.

$\int 2x + 3x^2 + 4x^3 \ dx$

Possible Answers:

F(x) =2 + 6x + 12x^2

F(x) =2x^2 + 3x^3 + 4x^4

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

Answer not listed.

F(x) =x^2 + x^3 + x^4

Correct answer:

F(x) =x^2 + x^3 + x^4

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 + 3x^2 + 4x^3 .

The antiderivative is F(x) =x^2 + x^3 + x^4 .

Report an Error

Example Question #58 : Indefinite Integrals

Evaluate.

$\int e^{3x- 2} \ dx$

Possible Answers:

$F(x) = \frac{e^{x}}{3}$

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

$F(x) = e^{3x- 2}$

Answer not listed.

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

Correct answer:

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

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, $f(x) = e^{3x- 2}$ .

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

Report an Error

Example Question #59 : Indefinite Integrals

Evaluate.

$\int 3x^2 + 12x^{11} \ dx$

Possible Answers:

Answer not listed

$F(x) = x^3 + x^{12}$

$F(x) = x^2 + x^{11}$

$F(x) = 3x^3 + 12x^{12}$

$F(x) = \frac{1}{3}x^3 +\frac{1}{12} x^{12}$

Correct answer:

$F(x) = x^3 + x^{12}$

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, $f(x) = 3x^2 + 12x^{11}$ .

The antiderivative is $F(x) = x^3 + x^{12}$ .

Report an Error

Example Question #60 : Indefinite Integrals

Evaluate.

$\int e^{4x- 2} \ dx$

Possible Answers:

$F(x) = e^{4x- 2}$

$F(x) = \frac{e^{x} }{4x-2}$

Answer not listed

$F(x) = \frac{e^{x} }{4}$

$F(x) = \frac{e^{4x- 2} }{4}$

Correct answer:

$F(x) = \frac{e^{4x- 2} }{4}$

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, $f(x) = e^{4x- 2}$ .

The antiderivative is $F(x) = \frac{e^{4x- 2} }{4}$ .

Report an Error

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Calculus 2 : Indefinite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #51 : Indefinite Integrals

Example Question #52 : Indefinite Integrals

Example Question #53 : Indefinite Integrals

Example Question #54 : Indefinite Integrals

Example Question #55 : Indefinite Integrals

Example Question #56 : Indefinite Integrals

Example Question #57 : Indefinite Integrals

Example Question #58 : Indefinite Integrals

Example Question #59 : Indefinite Integrals

Example Question #60 : Indefinite Integrals

All Calculus 2 Resources

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