Primitive function

9000071210

Level:

Evaluate the following integral on

R

\int (x + 2) \cos x d x

(x + 2) \sin x + \cos x + c, c \in R

- (\frac{x^{2}}{2} + 2 x) \sin x + c, c \in R

(x + 2) \sin x - \sin x + c, c \in R

9000071209

Level:

Evaluate the following integral on

R

\int \cos^{3} x \sin x d x

- \frac{\cos^{4} x}{4} + c, c \in R

- \frac{\sin^{4} x \cos x}{4} + c, c \in R

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

9000071204

Level:

Evaluate the following integral on the interval

(0; + \infty)

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

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

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

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

9000071205

Level:

Evaluate the following integral on

R

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

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

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

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

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

9000071203

Level:

Evaluate the following integral on the interval

(0; \frac{π}{2})

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

- 2 x - cotg x + c, c \in R

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

tg x - 2 x + c, c \in R

9000071207

Level:

Evaluate the following integral on the interval

(\sqrt{\frac{4}{3}}; + \infty)

\int \frac{6 x}{(3 x^{2} - 4)^{2}} d x

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

\frac{3 x^{2}}{x^{3} - 12 x^{2} + 16 x} + c, c \in R

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

9000071201

Level:

Evaluate the following integral on

R

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

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

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

6 x^{7} - 12 x^{4} + 4 x + c, c \in R

9000071202

Level:

Evaluate the following integral on the interval

(0; + \infty)

\int \frac{11 \sqrt{x^{3}} - 2}{\sqrt[3]{x^{2}}} d x

6 (x \sqrt[6]{x^{5}} - \sqrt[3]{x}) + c, c \in R

\frac{\frac{22}{5} \sqrt{x^{5}} - 2 x}{\frac{3}{5} \sqrt[3]{x^{5}}} + c, c \in R

\frac{121}{6} \sqrt[6]{x^{11}} - \frac{2}{3} \sqrt[3]{x} + c, c \in R

9000071206

Level:

Given the function

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

, find its primitive function

F

so that the graph of

F

passes through the point

A = [\frac{π}{2}; 3]

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

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

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

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