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Calculus 2 : Definite Integrals

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Example Questions

Example Question #85 : Finding Integrals

Evaluate.

$\int_{0}^{1}-\sin(6x) +3 \ dx$

Round to the nearest whole number.

Possible Answers:

Answer not listed.

Correct answer:

Explanation:

$\int_{a}^{b} f(x) dx$

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

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

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{1}-\sin(6x) +3 \ dx = \left ( \frac{\cos(6x)}{6} + 3x \right )_{0}^{1}$

$= \frac{\cos(6(1))}{6} + 3(1) - \left ( \frac{\cos(6(0))}{6} + 3(0) \right )$

= 3

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Example Question #86 : Finding Integrals

Evaluate.

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

Possible Answers:

Answer not listed.

4.25

-2.37

2.77

0.32

Correct answer:

2.77

Explanation:

$\int_{a}^{b} f(x) dx$

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

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

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{1}^{2} 2x - \frac{1}{3x} \ dx= \left ( x^2 - \frac{\ln(x)}{3} \right )_{1}^{2}$

$= 2^2 - \frac{\ln(2)}{3} - \left ( 1^2 - \frac{\ln(1)}{3} \right )$

= 2.77

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Example Question #87 : Finding Integrals

Evaluate.

$\int_{0}^{1} e^{7x} - 7 \ dx$

Possible Answers:

Answer not listed.

100.35

74.79

42.63

149.52

Correct answer:

149.52

Explanation:

$\int_{a}^{b} f(x) dx$

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

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

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{1} e^{7x} - 7 \ dx = \left ( \frac{e^{7x}}{7} - 7x \right )_{0}^{1}$

$= \frac{e^{7(1)}}{7} - 7(1) - \left ( \frac{e^{7(0)}}{7} - 7(0) \right )$

= 149.52

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Example Question #88 : Finding Integrals

Evaluate.

$\int_{1}^{2} \sin(x) + 6 - \frac{1}{5x}\ dx$

Possible Answers:

Answer not listed.

6.82

5.79

4.27

8.36

Correct answer:

6.82

Explanation:

$\int_{a}^{b} f(x) dx$

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

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

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{1}^{2} \sin(x) + 6 - \frac{1}{5x}\ dx = \left ( - \cos(x) + 6x - \frac{\ln (x)}{5} \right )_{1}^{2}$

$= - \cos(2) + 6(2) - \frac{\ln (2)}{5} - \left ( - \cos(1) + 6(1) - \frac{\ln (1)}{5} \right )$

= 6.82

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Example Question #89 : Finding Integrals

Evaluate.

$\int_{0}^{3} x^3 - \cos (2x) \ dx$

Possible Answers:

20.2

89.3

Answer not listed.

6.1

12.7

Correct answer:

20.2

Explanation:

In this case, $f(x) = x^3 - \cos (2x)$ .

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

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{3} x^3 - \cos (2x) \ dx = \left ( \frac{x^4}{4} - \frac{\sin (2x)}{2} \right )_{0}^{3}$

$= \frac{3^4}{4} - \frac{\sin (2(3))}{2} - \left ( \frac{0^4}{4} - \frac{\sin (2(0))}{2} \right )$

= 20.2

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Example Question #90 : Finding Integrals

Evaluate.

$\int_{0}^{1} 5e^{5x} \ dx$

Possible Answers:

12.5

102.4

67.2

147.4

Answer not listed.

Correct answer:

147.4

Explanation:

In this case, $f(x) = 5e^{5x}$ .

The antiderivative is $F(x) = e^{5x}$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{1} 5e^{5x} \ dx = \left ( e^{5x}\right )_{0}^{1}$

$= e^{5(1)} - \left ( e^{5(0)} \right )$

= 147.4

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Example Question #71 : Definite Integrals

Evaluate.

$\int_{0}^{1} \frac{1}{6x+5} \ dx$

Possible Answers:

4.64

0.98

Answer not listed.

0.13

1.35

Correct answer:

0.13

Explanation:

In this case, $f(x) = \frac{1}{6x+5}$ .

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

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{1} \frac{1}{6x+5} \ dx = \left ( \frac{\ln(6x+5)}{6} \right )_{0}^{1}$

$= \frac{\ln(6(1)+5)}{6} - \left ( \frac{\ln(6(0)+5)}{6} \right )$

= 0.13

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Example Question #72 : Definite Integrals

Evaluate.

$\int_{0}^{1} 3x - 5 \ dx$

Possible Answers:

Answer not listed.

-4.5

2.5

-1.5

-3.5

Correct answer:

-3.5

Explanation:

In this case, f(x) = 3x - 5 .

The antiderivative is $F(x) = \frac{3x^2}{2}-5x$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{1} 3x - 5 \ dx = \left ( \frac{3x^2}{2}-5x \right )_{0}^{1}$

$= \frac{3(1)^2}{2}-5(1) - \left ( \frac{3(0)^2}{2}-5(0) \right )$

=-3.5

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Example Question #73 : Definite Integrals

Evaluate.

$\int_{0}^{1} e^{8x} + \cos(4x) \ dx$

Possible Answers:

205.1

56.3

17.9

Answer not listed.

372.6

Correct answer:

372.6

Explanation:

In this case, $f(x) = e^{8x} + \cos(4x)$ .

The antiderivative is $F(x) =\frac{ \sin(4x)}{4} + \frac{e^{8x}}{8}$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{1} e^{8x} + \cos(4x) \ dx = \left ( \frac{ \sin(4x)}{4} + \frac{e^{8x}}{8} \right )_{0}^{1}$

$=\frac{ \sin(4(1))}{4} + \frac{e^{8(1)}}{8} - \left ( \frac{ \sin(4(0))}{4} + \frac{e^{8(0)}}{8} \right )$

= 372.6

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Example Question #74 : Definite Integrals

Evaluate.

$\int_{0}^{1} e^{9 - 2x}\ dx$

Possible Answers:

126.5

Answer not listed.

43.7

3503.2

352.4

Correct answer:

3503.2

Explanation:

$\int_{a}^{b} f(x) dx$

In this case, $f(x) = e^{9 - 2x}$ .

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

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{1} e^{9 - 2x}\ dx = \left ( -\frac{ e^{9 - 2x}}{2} \right )_{0}^{1}$

$= -\frac{ e^{9 - 2(1)}}{2} - \left ( -\frac{ e^{9 - 2(0)}}{2} \right )$

= 3503.2

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Calculus 2 : Definite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #85 : Finding Integrals

Example Question #86 : Finding Integrals

Example Question #87 : Finding Integrals

Example Question #88 : Finding Integrals

Example Question #89 : Finding Integrals

Example Question #90 : Finding Integrals

Example Question #71 : Definite Integrals

Example Question #72 : Definite Integrals

Example Question #73 : Definite Integrals

Example Question #74 : Definite Integrals

All Calculus 2 Resources

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