Primitive function

1003107906

Level: 
C
Solve the indefinite integral \[\int\frac{\mathrm{d}x}{\mathrm{cotg}\,x\cdot\sin2x} \] in the range \( \left(0;\frac{\pi}2\right) \).
\( \frac12\mathrm{tg}\,⁡x+c \), \( c\in\mathbb{R} \)
\( \frac12\mathrm{cotg}\,⁡x+c \), \( c\in\mathbb{R} \)
\( \ln⁡|\sin⁡2x |+c \), \( c\in\mathbb{R} \)
\( -\frac12\mathrm{tg}\,x+c \), \( c\in\mathbb{R} \)

1003107907

Level: 
C
Solve the indefinite integral \[ \int\mathrm{cotg}^2x\,\mathrm{d}x \] in the range \( (\pi; 2\pi) \).
\( -\mathrm{cotg}\,x-x+c \), \( c\in\mathbb{R} \)
\( \mathrm{cotg}\,x-x+c \), \( c\in\mathbb{R} \)
\( \mathrm{tg}\,x-x+c \), \( c\in\mathbb{R} \)
\( -\mathrm{tg}^2x+c \), \( c\in\mathbb{R} \)

1003107908

Level: 
C
Solve the indefinite integral \[ \int\frac{\cos⁡2x}{\sin^2⁡x}\mathrm{d}x \] in the range \( \left(\pi; \frac32\pi\right) \).
\( -\mathrm{cotg}\,x-2x+c \), \( c\in\mathbb{R} \)
\( \mathrm{cotg}\,x-2x+c \), \( c\in\mathbb{R} \)
\( -\mathrm{tg}\,x-2x+c \), \( c\in\mathbb{R} \)
\( -\mathrm{cotg}\,x+c \), \( c\in\mathbb{R} \)

1003107909

Level: 
C
Solve the indefinite integral \[ \int\frac{\sqrt[5]{\ln ⁡x}}x\mathrm{d}x \] in the range \( (0;\infty) \).
\( \frac56\ln⁡ x\cdot\sqrt[5]{\ln ⁡x}+c \), \( c\in\mathbb{R} \)
\( \frac56\ln ⁡x\cdot\sqrt[6]{\ln ⁡x}+c \), \( c\in\mathbb{R} \)
\( \frac65 \ln ⁡x\cdot\sqrt[5]{\ln ⁡x}+c \), \( c\in\mathbb{R} \)
\( \frac65\ln ⁡x\cdot\sqrt[6]{\ln ⁡x}+c \), \( c\in\mathbb{R} \)

1003107910

Level: 
C
Solve the indefinite integral \[ \int\mathrm{e}^{\sin ⁡x}\cos ⁡x\,\mathrm{d}x \] of a real-valued function.
\( \mathrm{e}^{\sin ⁡x} +c \), \( c\in\mathbb{R} \)
\( -\sin ⁡x\cdot\mathrm{e}^{\sin ⁡x} +c \), \( c\in\mathbb{R} \)
\( \mathrm{e}^{\cos ⁡x} +c \), \( c\in\mathbb{R} \)
\( \mathrm{e}^{\sin ⁡x}\cdot\cos ⁡x+c \), \( c\in\mathbb{R} \)

1003107911

Level: 
C
Solve the indefinite integral \[ \int\sin\sqrt x\,\mathrm{d}x \] in the range \( (0;\infty) \).
\( -2\sqrt x\cos\sqrt x+2\sin\sqrt x+c \), \( c\in\mathbb{R} \)
\( 2\sqrt x\cos⁡\sqrt x+2\sin\sqrt x+c \), \( c\in\mathbb{R} \)
\( 2\sqrt x\cos\sqrt x-2\sin\sqrt x+c \), \( c\in\mathbb{R} \)
\( -\cos\sqrt x \), \( c\in\mathbb{R} \)

1003107912

Level: 
C
Which method is the most effective to solve the indefinite integral \[ \int\frac{\mathrm{d}x}{x\ln ⁡x} \] in the range \( (1;\infty) \)?
By substitution, \( a=\ln ⁡x \).
By parts integration, when we let \( u(x)=\frac1x \), where \( u(x) \) is the integrated function, and we let \( v'(x)=\ln ⁡x \), where \( v'(x) \) is the differentiated function.
By substitution, \( a=\frac1x \).
By factorization into \( \int\frac1x\mathrm{d}x\cdot\int\frac1{\ln ⁡x}\mathrm{d}x \).

1003107913

Level: 
C
Which method is the most effective to solve the indefinite integral \[ \int\sin(\ln x)\mathrm{d}x \] in the range \( (0;\infty) \)?
By parts integration, when we let \( u(x)=\sin⁡(\ln ⁡x) \), where \( u(x) \) is the integrated function, and we let \( v'(x)=1 \), where \( v'(x) \) is the differentiated function.
By substitution, \( a=\sin ⁡x \).
By parts integration, when we let \( u(x)=\ln x \), where \( u(x) \) is the integrated function, and we let \( v'(x)=\sin x \), where \( v'(x) \) is the differentiated function.
By substitution, \( t=\sin⁡(\ln⁡ x) \).

2010000304

Level: 
C
Solve the indefinite integral \[ \int\mathrm{e}^{\cos ⁡x}\sin ⁡x\,\mathrm{d}x \] of a real-valued function.
\( -\mathrm{e}^{\cos ⁡x} +c \), \( c\in\mathbb{R} \)
\(- \mathrm{e}^{\cos ⁡x}\cdot\cos ⁡x+c \), \( c\in\mathbb{R} \)
\( \mathrm{e}^{\sin ⁡x}\cdot\cos ⁡x+c \), \( c\in\mathbb{R} \)
\( \mathrm{e}^{\cos ⁡x}\cdot\sin ⁡x+c \), \( c\in\mathbb{R} \)

2010000305

Level: 
C
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int \log_2 x\, \mathrm{d}x \]
\(x\log_2x -\frac{x} {\ln 2}+ c,\ c\in \mathbb{R}\)
\(\log_2 x -\frac{x} {\ln 2}+ c,\ c\in \mathbb{R}\)
\(x\log_2 x -x+ c,\ c\in \mathbb{R}\)
\(x\log_2 x +\frac{x} {\ln 2}+ c,\ c\in \mathbb{R}\)