9000150401
Level:
A
Evaluate the following definite integral.
\[
\int _{-3}^{1}(x^{2} + 3x)\, \text{d}x
\]
\(-\frac{8}
{3}\)
\(\frac{8}
{3}\)
\(-\frac{64}
{3} \)
\(\frac{64}
{3} \)
A
9000150402
Level:
A
Evaluate the following definite integral.
\[
\int _{-\frac{\pi }{
2} }^{ \frac{\pi }
{2} }\sin x\, \text{d}x
\]
\(0\)
\(\pi \)
\(2\)
\(1\)
9000150403
Level:
A
Evaluate the following definite integral.
\[
\int _{-2}^{0}\mathrm{e}^{x}\, \text{d}x
\]
\(1 -\frac{1}
{\mathrm{e}^{2}} \)
\(1 + \frac{1}
{\mathrm{e}^{2}} \)
\(\frac{1}
{\mathrm{e}^{2}} \)
\(-\frac{1}
{\mathrm{e}^{2}} \)
9000150404
Level:
A
Evaluate the following definite integral.
\[
\int _{2}^{6} \frac{2}
{x}\, \text{d}x
\]
\(\ln 9\)
\(\ln 2\)
\(\ln 3\)
\(2\ln 6\)
9000150407
Level:
A
Evaluate the following definite integral.
\[
\int _{1}^{2}7^{x}\, \text{d}x
\]
\(\frac{42}
{\ln 7} \)
\(49\ln 7\)
\(42\)
\(42\ln 7\)
9000150408
Level:
A
Evaluate the following definite integral.
\[
\int _{0}^{ \frac{\pi }{
4} } \frac{2}
{\cos ^{2}x}\, \text{d}x
\]
\(2\)
\(0\)
\(4\)
\(\pi \)
9000151310
Level:
A
Find the value of the real parameter
\(a\) which ensures, that
the following two lines \(p\)
and \(q\)
are perpendicular.
\[
p\colon ax + y - 4 = 0,\qquad q\colon x + 2y + 4 = 0.
\]
\(- 2\)
\(2\)
\(1\)
\(- 1\)
9000150105
Level:
A
Evaluate the following integral on \(\mathbb{R}\).
\[
\int \left (6^{x} - 6x^{6}\right )\, \mathrm{d}x
\]
\(\frac{6^{x}}
{\ln 6} -\frac{6x^{7}}
{7} + c,\ c\in \mathbb{R}\)
\(6^{x}\ln 6 - 6x^{7} + c,\ c\in \mathbb{R}\)
\(6^{x}\ln 6 -\frac{6x^{7}}
{7} + c,\ c\in \mathbb{R}\)
\(\frac{6^{x}}
{\ln 6} - 6x^{7} + c,\ c\in \mathbb{R}\)
9000150108
Level:
A
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int \left (\frac{3}
{x} - 3x^{-2} + \frac{2}
{x^{3}}\right )\, \mathrm{d}x
\]
\(3\ln |x| + \frac{3}
{x} - \frac{1}
{x^{2}} + c,\ c\in \mathbb{R}\)
\(3\ln |x|-\frac{3}
{x} - \frac{1}
{x^{2}} + c,\ c\in \mathbb{R}\)
\(3\ln |x| + \frac{3}
{x} + \frac{1}
{x^{2}} + c,\ c\in \mathbb{R}\)
\(3\ln |x|-\frac{3}
{x} + \frac{1}
{x^{2}} + c,\ c\in \mathbb{R}\)
9000150301
Level:
A
Evaluate the following integral on \(\mathbb{R}\).
\[
\int 9x^{8}\, \text{d}x
\]
\(x^{9} + c,\ c\in \mathbb{R}\)
\(9x^{9} + c,\ c\in \mathbb{R}\)
\(\frac{x^{9}}
{9} + c,\ c\in \mathbb{R}\)
\(9x^{7} + c,\ c\in \mathbb{R}\)