Primitive Function

9000071207

Level: 
B
Evaluate the following integral on the interval \(\left(\sqrt{\frac43},+\infty\right)\). \[ \int \frac{6x} {(3x^{2} - 4)^{2}}\, \mathrm{d}x \]
\(\frac{1} {4-3x^{2}} + c,\ c\in \mathbb{R}\)
\(\frac{3x^{2}} {x^{3}-12x^{2}+16x} + c,\ c\in \mathbb{R}\)
\(\frac{1} {(3x^{2}-4)^{2}} + c,\ c\in \mathbb{R}\)

9000071208

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

9000066009

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

9000066010

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

9000066008

Level: 
B
Evaluate the following integral on \(\mathbb{R}\). \[ \int x\mathrm{e}^{x}\, \mathrm{d}x \]
\(x\mathrm{e}^{x} -\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(x^{2}\mathrm{e}^{x} - 2x\mathrm{e}^{x} + 2\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(2x^{3}\mathrm{e}^{x} - x\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(\frac{1} {2}x^{2}\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)

9000065901

Level: 
B
Evaluate the following integral on the interval \((-1,+\infty)\). \[ \int \frac{1} {x + 1}\, \text{d}x \]
\(\ln |x + 1| + c,\ c\in \mathbb{R}\)
\(\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{1} {x} + c,\ c\in \mathbb{R}\)
\(-\frac{1} {2}(x + 1)^{-2} + c,\ c\in \mathbb{R}\)

9000065902

Level: 
A
Evaluate the following integral on the interval \((0,+\infty)\). \[ \int \left (2 + \frac{1} {x}\right )\, \text{d}x \]
\(2x +\ln |x| + c,\ c\in \mathbb{R}\)
\(\ln |x| + c,\ c\in \mathbb{R}\)
\(2 +\ln |x| + c,\ c\in \mathbb{R}\)
\(2x^{2} +\ln |x| + c,\ c\in \mathbb{R}\)

9000065903

Level: 
B
Evaluate the following integral on the interval \((-6,+\infty)\). \[ \int \frac{1} {6x + 36}\, \text{d}x \]
\(\frac{1} {6}\ln |x + 6| + c,\ c\in \mathbb{R}\)
\(-\frac{1} {2}(6x + 36)^{-2} + c,\ c\in \mathbb{R}\)
\(6\ln |x + 6| + c,\ c\in \mathbb{R}\)
\(12x^{2} + 36x + c,\ c\in \mathbb{R}\)

9000065904

Level: 
B
Evaluate the following integral on the interval \((0,+\infty)\). \[ \int \frac{x^{3} + 2x} {x^{2}} \, \text{d}x \]
\(\frac{1} {2}x^{2} + 2\ln |x| + c,\ c\in \mathbb{R}\)
\(x +\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{1} {4}x^{4} + 4x^{2} +\ln |x^{2}| + c,\ c\in \mathbb{R}\)
\(2x^{2} + 2 +\ln |x^{2}| + c,\ c\in \mathbb{R}\)