Primitive function

1003107911

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

1003107910

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

1003107909

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

1003107908

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

1003107907

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

1003107906

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

1003107905

Level: 
C
Solve the indefinite integral \[ \int\left(ab\mathrm{e}^c-bx^2+5^b-\sin ⁡c\right) \mathrm{d}c \] of a real-valued function, where \( a \), \( b \), \( x \) are real numbers.
\( ab\mathrm{e}^c-bcx^2+c\cdot5^b+\cos ⁡c+k \), \( k\in\mathbb{R} \)
\( \mathrm{e}^c-\cos ⁡c+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c-bcx^2+c\cdot5^b-\cos ⁡c+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c-bcx^2+5^b\cdot c-\sin ⁡c+k \), \( k\in\mathbb{R} \)

1003107904

Level: 
C
Solve the indefinite integral \[ \int\left(ab\mathrm{e}^c-bx^2+5^b-\sin ⁡c\right) \mathrm{d}b \] of a real-valued function, where \( x \), \( a \), \( c \) are real numbers.
\( 0.5a\mathrm{e}^cb^2-\frac{b^2}2 x^2+\frac{5^b}{\ln⁡5} -b\sin ⁡c+k \), \( k\in\mathbb{R} \)
\( 0.5a\mathrm{e}^cb^2-\frac{b^2}2\cdot\frac{x^3}3+\frac{5^b}{\ln⁡5} -b \sin⁡ c+k \), \( k\in\mathbb{R} \)
\( a\mathrm{e}^c-2bx+5^b-\sin ⁡c+k \), \( k\in\mathbb{R} \)
\( a\mathrm{e}^c-bx^2+5^b-b\sin c+k \), \( k\in\mathbb{R} \)

1003107903

Level: 
C
Solve the indefinite integral \[ \int\left( ab\mathrm{e}^c-bx^2+5^b-\sin ⁡c\right)\mathrm{d}a \] of a real-valued function, where \( x \), \( b \), \( c \) are real numbers.
\( \frac{a^2 b\mathrm{e}^c}2-abx^2+a5^b-a\sin⁡ c+k \), \( k\in\mathbb{R} \)
\( \frac{a^2}2b\mathrm{e}^c+k \), \( k\in\mathbb{R} \)
\( b\mathrm{e}^c-bx^2+5^b-\sin⁡ c+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c-b\frac{x^3}3+k \), \( k\in\mathbb{R} \)

1003107902

Level: 
C
Solve the indefinite integral \[ \int\left(ab\mathrm{e}^c-bx^2+5^b-\sin ⁡c \right)\mathrm{d}x \] of a real-valued function, where \( a \), \( b \), \( c \) are real numbers.
\( ab\mathrm{e}^c x-b\frac{x^3}3+5^b x-x \sin c+k \), \( k\in\mathbb{R} \)
\( -b\frac{x^3}3+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c-2b+5^b-\sin c+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c x-2bx+5^b x-x \sin c+k \), \( k\in\mathbb{R} \)