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9000071208

Level:

Evaluate the following integral on the interval

(0; + \infty)

\int x^{2} \ln x d x

\frac{x^{3}}{3} (\ln x - \frac{1}{3}) + c, c \in R

\frac{x^{2}}{3} + c, c \in R

x^{2} (\frac{x \ln x}{3} - \frac{1}{2}) + c, c \in R

9000070508

Level:

The sum of the first three terms of a geometric sequence is

27

. The sum of the next three terms is

512

. Find the quotient.

\frac{8}{3}

2

3

\frac{3}{2}

\frac{4}{3}

9000070504

Level:

The geometric sequence is formed by consecutive powers of the number

3

. The eighth term is

a_{8} = 3^{10}

. Find the sum of the first five terms.

3 267

1 089

2 178

4 356

6 543

9000070507

Level:

A car depreciates (loses value) at the rate of

15 %

per year. After how many full years will the value of the car decrease to less than one quarter of its initial value?

9

6

7

8

10

9000070503

Level:

Three numbers form a geometric sequence. The sum of these numbers is

39

and the product is

1 000

. Find the smallest of these numbers.

4

2

2.5

10

25

9000070505

Level:

The dimensions of the box form a geometric sequence. The volume of the box is

27 {cm}^{3}

and the length of the shortest side is

2 cm

. Find the surface area of the box.

57 {cm}^{2}

28.5 {cm}^{2}

27 {cm}^{2}

35 {cm}^{2}

45 {cm}^{2}

9000070506

Level:

The intensity of the light is reduced by

8 %

when the light shines through the glass panel. How much of the light intensity remains if the light shines through

6

glass panels?

60.6 %

39.4 %

52 %

48 %

3.14 %

9000066001

Level:

Identify the formula for integration by parts.

\int u (x) v^{'} (x) d x = u (x) v (x) - \int u^{'} (x) v (x) d x

\int u (x) v (x) d x = u^{'} (x) v^{'} (x) - \int u^{'} (x) v (x) d x

\int u^{'} (x) v^{'} (x) d x = u (x) v (x) - \int u^{'} (x) v (x) d x

\int u (x) v^{'} (x) d x = u (x) v (x) + \int u^{'} (x) v (x) d x

9000066004

Level:

Evaluate the following integral on

R

\int x^{2} \sin x d x

- x^{2} \cos x + 2 x \sin x + 2 \cos x + c, c \in R

x^{2} \cos x - 2 x \sin x - 2 \cos x + c, c \in R

\frac{1}{3} x^{3} \cos x + c, c \in R

\frac{1}{3} x^{3} - \cos x + c, c \in R

9000066006

Level:

Evaluate the following integral on the interval

(0; + \infty)

\int x \ln x d x

\frac{1}{2} x^{2} \ln x - \frac{1}{4} x^{2} + c, c \in R

x \ln x - \frac{1}{2} x^{2} + c, c \in R

x \ln x - x + c, c \in R

\frac{1}{2} x^{2} + \frac{1}{| x |} + c, c \in R

C

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