Definite integral

1003108203

Level:

Compare the definite integral

I = \int_{0}^{\frac{π}{4}} \frac{\cos 2 b}{\cos^{2} b} d b

to the number

\frac{π}{2}

I

is smaller than

\frac{π}{2}

1

I

is bigger than

\frac{π}{2}

1

I

is equal to

\frac{π}{2}

I

is smaller than

\frac{π}{2}

\frac{π}{4}

1003108204

Level:

The value of the definite integral

\int_{- \frac{π}{4}}^{\frac{π}{4}} ({tg}^{2} x + 1) d x

is:

a whole number

a decimal number

a proper fraction

an irrational number

1003108205

Level:

Compare the two definite integrals

I_{1} = \int_{- 1}^{1} (x + \frac{π}{2}) d x

and

I_{2} = \int_{0}^{\frac{π}{4}} tg x \cdot \cos x d x

I_{1}

is bigger than

I_{2}

I_{1}

is smaller than

I_{2}

I_{1}

is equal to

I_{2}

These integrals cannot be compared.

1003118801

Level:

Which of the following formulas is not equal to

\int_{4}^{8} \frac{3 x + 1}{x^{2} - x - 6} d x

\int_{4}^{8} \frac{2}{x - 3} d x - \int_{4}^{8} \frac{1}{x + 2} d x

\int_{4}^{8} \frac{2}{x - 3} d x + \int_{4}^{8} \frac{1}{x + 2} d x

\int_{4}^{6} \frac{3 x + 1}{x^{2} - x - 6} d x + \int_{6}^{8} \frac{3 x + 1}{x^{2} - 6 - x} d x

\int_{8}^{4} \frac{- 1 - 3 x}{x^{2} - x - 6} d x

1003118802

Level:

Find positive real numbers

a

and

b

a

b > 0

, such that

b - a = 6

and

\int_{a}^{b} (4 x - 6) d x = 60

a = 1

b = 7

a = 7

b = 1

a = 24

b = 30

a = 1.5

b = 7.5

1003118804

Level:

Evaluate the integral

\int_{- 3}^{2} f (x) d x

, where

f (x) = x^{- 4}

for

x \in [- 3; - \frac{1}{2}]

and

f (x) = 12.8 - 6.4 x

for

x \in [- \frac{1}{2}; 2]

. (rounded to

2

decimal places)

22.65

43. \overset{―}{8}

44. \overset{―}{1}

29.7 \overset{―}{1}

1003118805

Level:

Evaluate the integral

\int_{- 5}^{5} f (x) d x

, where

f (x) = x^{2} + 2

for

x \in [- 5; 1]

and

f (x) = 3

for

x \in [1; 5]

66

\frac{7}{3}

\frac{25}{3}

42

1003118901

Level:

Evaluate the definite integral.

\int_{- 2}^{3} \sqrt{5 x + 12} d x

18.3

19.1

18.1

19.3

1003118902

Level:

Evaluate the definite integral.

\int_{0}^{10} \frac{4 x}{x^{2} + 1} d x

\ln (10 201)

\ln (10 404)

\ln (202)

9

1003118903

Level:

Evaluate the definite integral.

\int_{- 0.5}^{0.1} e^{5 x - 1} d x

0.115

0.127

0.121

0.205

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