Definite integral

9000150406

Level:

Evaluate the following definite integral.

\int_{- 3}^{2} \frac{x}{x^{2} + 3} d x

\frac{1}{2} \ln \frac{7}{12}

2 \ln \frac{12}{7}

2 \ln \frac{7}{12}

\frac{1}{2} \ln \frac{12}{7}

1003027501

Level:

Evaluate the following definite integral.

\int_{e}^{e^{3}} \frac{10 x - 14}{x \cdot (x - 2)} d x

3 \ln \frac{e^{3} - 2}{e - 2} + 14

3 \ln [(e^{3} - 2) (e - 2)] + 14

\ln \frac{e^{3} - 2}{e - 2} + 14

3 \ln (e^{3} - 2) + \ln (e - 2)^{3} + 14

1003027502

Level:

Evaluate the following definite integral.

\int_{5}^{25} \frac{4 x + 1}{x^{2} - x - 2} d x

\ln \frac{13}{3} + 3 \ln \frac{23}{3}

\ln (26 \cdot 23^{3} \cdot 6 \cdot 3^{3})

\ln \frac{26}{27} + 3 \ln 23 + \ln 6

\ln (24 \cdot 27^{3}) - \ln (4 \cdot 7^{3})

1003027503

Level:

Evaluate the following definite integral.

\int_{e}^{12} \frac{x + 5}{\frac{1}{4} x \cdot (x + 4)} d x

\ln (e + 4) + 5 \ln 12 - 2 \ln 4 - 5

5 + 2 \ln 4 + 5 \ln 12 - \ln (e + 4)

\ln \frac{15}{4} - 5 + \ln (4 + e)

\frac{\ln \frac{12^{5}}{16}}{\ln \frac{e^{5}}{e + 4}}

1003118905

Level:

Evaluate the definite integral.

\int_{e}^{e^{2}} \frac{\ln x}{x} d x

1.5

2.5

e^{2}

e^{2} - e

1003118906

Level:

Evaluate the definite integral.

\int_{1}^{e} (\sqrt[3]{x} - x^{3}) \cdot \ln x d x

- 9.03

9.03

11.4

- 11.4

1003124302

Level:

Which of the given values of a real number

a \in (\frac{π}{2}; π)

makes the equality

\int_{a}^{2 a} (3 \sin x - 4 x) d x = - 6 a^{2}

true?

a = \frac{2}{3} π

a = \frac{5}{6} π

a = \frac{3}{4} π

a = \frac{7}{8} π

1003124303

Level:

Which of the given values of real numbers

a

b \in (0; \frac{π}{2})

, such as

a < b

, makes the equality

\int_{a}^{b} \cos x d x = 2 \cos \frac{π}{4} \cdot \sin \frac{π}{12}

true?

a = \frac{π}{6}

b = \frac{π}{3}

a = \frac{π}{3}

b = \frac{π}{6}

a = \frac{π}{3}

b = \frac{π}{4}

a = \frac{π}{4}

b = \frac{π}{3}

1003124304

Level:

Given a function

f (x) = a x^{4} + b x

, find real numbers

a

and

b

, such that

\int_{0}^{1} f (x) d x = 27

and

\int_{- 1}^{0} f (x) d x = 57

a = 210

b = - 30

a = 210

b = 30

a = 75

b = 60

a = 30

b = 210

1003124305

Level:

Given a function

f (x) = a x^{6} + b x^{3} + c x + 8

, find real numbers

a

b

and

c

, such that

\int_{0}^{1} f (x) d x = \frac{35}{4}

f^{'} (0) = 2

and

f^{'} (1) = 180

a = 7

b = - 5

c = 2

a = 7

b = 5

c = 2

a = - 7

b = - 5

c = 2

a = - 7

b = 5

c = - 2

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9000150406

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1003124303

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1003124305

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