Primitive function

2010005102

Level:

Evaluate the following integral on

(\frac{π}{2}; π)

\int (5 \sin x - \frac{3}{\cos^{2} x} - \frac{7}{\sin^{2} x}) d x

- 5 \cos x - 3 tg x + 7 cotg x + c, c \in R

5 \cos x + 3 tg x + 7 cotg x + c, c \in R

- 5 \cos x - 3 tg x - 7 cotg x + c, c \in R

5 \cos x + 3 tg x + 7 cotg x + c, c \in R

2010005103

Level:

Evaluate the following integral on

(0; \infty)

\int (6 \sqrt{x} - 5 \sqrt[3]{x^{2}} + 10 \sqrt[4]{x}) d x

4 x \sqrt{x} - 3 x \sqrt[3]{x^{2}} + 8 x \sqrt[4]{x} + c, c \in R

9 x \sqrt{x} - \frac{25}{3} x^{3} \sqrt[3]{x^{2}} + \frac{25}{2} x \sqrt[4]{x} + c, c \in R

4 \sqrt{x} - 3 \sqrt[3]{x^{2}} + 8 \sqrt[4]{x} + c, c \in R

x + 20 \sqrt{x} + c, c \in R

2010005104

Level:

Evaluate the following integral on the interval

(0; + \infty)

\int (2 x^{- 1} + \frac{2}{x^{2}} - 3 x^{- 3}) d x

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

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

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

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

2010005105

Level:

Given the function

f (x) = \cos x - \sin x

, find its primitive function F so that the graph of

F

passes through the point

A = [π; 3]

F (x) = \sin x + \cos x + 4

F (x) = \cos x - \sin x + 4

F (x) = \sin x - \cos x + 2

2110005106

Level:

In which picture can you see a pair of graphs of functions

f_{1}

and

f_{2}

that are primitive to the same function?

9000065501

Level:

Evaluate the following integral on

R

\int (x^{3} + x^{2} - 2 x) d x

\frac{1}{4} x^{4} + \frac{1}{3} x^{3} - x^{2} + c, c \in R

\frac{1}{4} x^{4} - \frac{1}{3} x^{3} + x^{2} + c, c \in R

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

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

9000065502

Level:

Evaluate the following integral on

R

\int (4 x + 7) d x

2 x^{2} + 7 x + c, c \in R

2 x^{2} - 7 x + c, c \in R

4 + c, c \in R

4 x^{2} + 7 x + c, c \in R

9000065503

Level:

Evaluate the following integral on the interval

(0; + \infty)

\int (4 x^{- 3} - x^{- 4}) d x

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

- \frac{4}{3} x^{- 2} - \frac{1}{3} x^{- 3} + c, c \in R

- \frac{3}{4} x^{- 4} - \frac{1}{5} x^{- 5} + c, c \in R

- 12 x^{2} + 4 x^{- 3} + c, c \in R

9000065507

Level:

Given the function

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

find the function

f

such that

F

is primitive to

f

R

f (x) = x^{3} - 2 x^{2}

f (x) = x^{5} - 2 x^{4}

f (x) = x^{5} - 3 x^{2}

f (x) = - 4 x^{- 4} - 3 x^{2}

9000065508

Level:

Given the function

F (x) = \frac{1}{4} x^{4} - \frac{5}{2} x^{2},

find the function

f

such that

F

is primitive to

f

R

f (x) = x (x^{2} - 5)

f (x) = x^{3} - 5 x^{2}

f (x) = x^{5} - 5 x^{3}

f (x) = x^{5} - 2 x

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