Primitive function

1003107601

Level: 
C
Evaluate the following integral on \( (3;\infty) \). \[ \int\frac{5x-3}{x^2-2x-3}\mathrm{d}x \]
\( \ln\!⁡\left[(x-3)^3\cdot(x+1)^2\right]+c\text{, }c\in\mathbb{R} \)
\( \ln\!⁡\left[(x-3)^2\cdot(x+1)^3\right]+c\text{, }c\in\mathbb{R} \)
\( \ln\frac{(x-3)^2}{(x+1)^3}+c\text{, }c\in\mathbb{R} \)
\( \ln\frac{(x-3)^3}{(x+1)^2}+c\text{, }c\in\mathbb{R} \)

1003107602

Level: 
C
Evaluate the following integral on \( (3;\infty) \). \[ \int\frac7{x^2+x-12}\mathrm{d}x \]
\( \ln\frac{x-3}{x+4}+c\text{, }c\in\mathbb{R} \)
\( \ln\!\left[(x-3)(x+4)\right]+c\text{, }c\in\mathbb{R} \)
\( \ln⁡\frac{x+4}{x-3}+c\text{, }c\in\mathbb{R} \)
\( \ln\frac{x+6}{x-2}+c\text{, }c\in\mathbb{R} \)

1003107603

Level: 
C
Evaluate the following integral on \( (0;\infty) \). \[ \int\frac{3x^3+3x^2-x+1}{x^2+x}\mathrm{d}x \]
\( 1.5x^2+\ln\frac x{x^2+2x+1}+c\text{, }c\in\mathbb{R} \)
\( \ln\frac x{(x+1)^2}+3x+c\text{, }c\in\mathbb{R} \)
\( \ln\!⁡\left[x\cdot(x+1)^2\right]+3x+c\text{, }c\in\mathbb{R} \)
\( \ln\frac{(x+1)^2}x+c\text{, }c\in\mathbb{R} \)

1003107604

Level: 
C
Evaluate the following integral on \( (0;\infty) \). \[ \int\frac{-2x^2+3x+2}{x^3+x^2}\mathrm{d}x \]
\( \ln\frac x{x^3+3x^2+3x+1}-\frac2x+c\text{, }c\in\mathbb{R} \)
\( \ln x-\ln(x+1)^3+c\text{, }c\in\mathbb{R} \)
\( \ln\frac{x\cdot x^2}{(x+1)^3}+c\text{, }c\in\mathbb{R} \)
\(\ln\frac{x^3}{x^3+3x^2+3x+1}-\frac2x+c\text{, }c\in\mathbb{R} \)

1003107605

Level: 
C
Evaluate the following integral on \( (0;\infty) \). \[ \int\frac{7x+2}{15x^2+3x}\mathrm{d}x \]
\( \frac23\ln⁡ x-\frac15\ln\left(x+\frac15\right)+c\text{, }c\in\mathbb{R} \)
\( \ln\frac{\sqrt[3]{x^2}}{\sqrt[5]{x+1}}+c\text{, }c\in\mathbb{R} \)
\( \frac23\ln x-\ln(5x+1)+c\text{, }c\in\mathbb{R} \)
\(\ln\frac{\sqrt[3]{x^2}}{\sqrt[5]{x+5}}+c\text{, }c\in\mathbb{R} \)

1003107901

Level: 
C
Solve the indefinite integral \[ \int\sin^3 x\cos^2x\,\mathrm{d}x \] of a real-valued function using a suitable substitution.
\( \frac{\cos^5⁡x}5-\frac{\cos^3⁡x}3+c \)
\( -\frac{\cos^5⁡x}5+\frac{\cos^3⁡x}3+c \)
\( \frac{\cos^5⁡x}5+\frac{\cos^3⁡x}3+c \)
\( -\frac{\cos^5⁡x}5-\frac{\cos^3⁡x}3+c \)

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.
\( ab\mathrm{e}^c x-b\frac{x^3}3+5^b x-x \sin c+k \), \( k\in\mathbb{R} \)
\( -b\frac{x^3}3+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c-2b+5^b-\sin c+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c x-2bx+5^b x-x \sin c+k \), \( k\in\mathbb{R} \)

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.
\( \frac{a^2 b\mathrm{e}^c}2-abx^2+a5^b-a\sin⁡ c+k \), \( k\in\mathbb{R} \)
\( \frac{a^2}2b\mathrm{e}^c+k \), \( k\in\mathbb{R} \)
\( b\mathrm{e}^c-bx^2+5^b-\sin⁡ c+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c-b\frac{x^3}3+k \), \( k\in\mathbb{R} \)

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.
\( 0.5a\mathrm{e}^cb^2-\frac{b^2}2 x^2+\frac{5^b}{\ln⁡5} -b\sin ⁡c+k \), \( k\in\mathbb{R} \)
\( 0.5a\mathrm{e}^cb^2-\frac{b^2}2\cdot\frac{x^3}3+\frac{5^b}{\ln⁡5} -b \sin⁡ c+k \), \( k\in\mathbb{R} \)
\( a\mathrm{e}^c-2bx+5^b-\sin ⁡c+k \), \( k\in\mathbb{R} \)
\( a\mathrm{e}^c-bx^2+5^b-b\sin c+k \), \( k\in\mathbb{R} \)

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.
\( ab\mathrm{e}^c-bcx^2+c\cdot5^b+\cos ⁡c+k \), \( k\in\mathbb{R} \)
\( \mathrm{e}^c-\cos ⁡c+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c-bcx^2+c\cdot5^b-\cos ⁡c+k \), \( k\in\mathbb{R} \)
\( ab\mathrm{e}^c-bcx^2+5^b\cdot c-\sin ⁡c+k \), \( k\in\mathbb{R} \)