Primitive function

9000065509

Level: 
A
Given the function \[ F(x) = x + \frac{9} {2}x^{2} + 9x^{3} + \frac{27} {4} x^{4}, \] find the function \(f\) such that \(F\) is primitive to \(f\) on \(\mathbb{R}\).
\(f(x) = (1 + 3x)^{3}\)
\(f(x) = (1 + 3x)^{2}\)
\(f(x) = 1 + 3x + 3x^{2} + 3x^{3}\)
\(f(x) = (1 + 3x)^{4}\)

9000065902

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

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 \((0;+\infty )\).
\(f(x) = \frac{x^{2}+2x+1} {x^{2}} \)
\(f(x) = \frac{x^{2}} {(x+1)^{2}} \)
\(f(x) = \frac{x^{2}-1} {x^{2}} \)
\(f(x) = \frac{x^{2}} {(x-1)^{2}} \)

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 \(\mathbb{R}\). \[ \int \left (x^{2} + 2^{x}\right )\, \mathrm{d}x \]
\(\frac{x^{3}} {3} + \frac{2^{x}} {\ln 2} + c,\ c\in \mathbb{R}\)
\(\frac{x^{3}} {3} + \frac{2^{x+1}} {x+1} + c,\ c\in \mathbb{R}\)
\(2x + \frac{2^{x}} {\ln \left |x\right |} + c,\ c\in \mathbb{R}\)

9000150101

Level: 
A
Evaluate the following integral on \(\mathbb{R}\). \[ \int \left (\cos x -\sin x\right )\, \mathrm{d}x \]
\(\sin x +\cos x + c,\ c\in \mathbb{R}\)
\(\sin x -\cos x + c,\ c\in \mathbb{R}\)
\(-\sin x +\cos x + c,\ c\in \mathbb{R}\)
\(-\sin x -\cos x + c,\ c\in \mathbb{R}\)