Primitive function

9000071204

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

9000071205

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

9000065908

Level: 
A
Given the function \[ F(x) = \frac{1} {2}x^{2} - x, \] find the function \(f\) such that \(F\) is primitive to \(f\) on \((1;+\infty )\).
\(f(x) = \frac{x^{2}-1} {x+1} \)
\(f(x) = \frac{x^{2}-1} {x-1} \)
\(f(x) = \frac{x+1} {x^{2}-1}\)
\(f(x) = \frac{x-1} {x^{2}-1}\)

9000065910

Level: 
A
Given function \[ F(x) = x + 2\ln |x|-\frac{1} {x}, \] find the function \(f\) such that \(F\) is primitive to \(f\) on \((0;+\infty )\).
\(f(x) = \frac{x^{2}+2x+1} {x^{2}} \)
\(f(x) = \frac{x^{2}} {(x+1)^{2}} \)
\(f(x) = \frac{x^{2}-1} {x^{2}} \)
\(f(x) = \frac{x^{2}} {(x-1)^{2}} \)

9000066001

Level: 
C
Identify the formula for integration by parts.
\(\int u(x)v'(x)\, \mathrm{d}x = u(x)v(x) -\int u'(x)v(x)\, \mathrm{d}x\)
\(\int u(x)v(x)\, \mathrm{d}x = u'(x)v'(x) -\int u'(x)v(x)\, \mathrm{d}x\)
\(\int u'(x)v'(x)\, \mathrm{d}x = u(x)v(x) -\int u'(x)v(x)\, \mathrm{d}x\)
\(\int u(x)v'(x)\, \mathrm{d}x = u(x)v(x) +\int u'(x)v(x)\, \mathrm{d}x\)

9000066004

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

9000066006

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

9000066009

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