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Integrals

Multiple Choice Questions

171.

$\int \frac{\cos (x) - \sin (x)}{1 + 2 \sin (x) \cos (x)} dx$ is equal to

$- \frac{1}{\cos (x) - \sin (x)} + c$
$\frac{\cos (x) + \sin (x)}{\cos (x) - \sin (x)} + c$
$- \frac{1}{\sin (x) + \cos (x)}$
$\frac{x}{\sin (x) + \cos (x)} + c$

172.

$\int \frac{1}{x} \log_{ex} (e) dx$ is equal to

$\log_{e} (1 - \log_{e} (x)) + c$
$\log_{e} (\log_{e} (ex) - 1)$ + c
$\log_{e} (\log_{e} (x) - 1)$ + c
$\log_{e} (1 + \log_{e} (x)) + c$

173.
The value of $\int_{1}^{e} 10^{\log_{e} (x)} d x$ is equal to

10 $\log_{e} (10 e)$

$\frac{10 e - 1}{\log_{e} (10 e)}$

$\frac{10 e}{\log_{e} (10 e)}$

$(10 e) \log_{e} (10 e)$

$\frac{10 e - 1}{\log_{e} (10 e)}$

$Let I = \int_{1}^{e} 10^{\log_{e} (x)} dx Again, let I_{1} = \int 10^{\log_{e} (x)} dx \Rightarrow I_{1} = x . 10^{\log_{e} (x)} - \int x . 10^{\log_{e} (x)} . \frac{\log_{e} (10)}{x} dx \Rightarrow I_{1} = x . 10^{\log_{e} (x)} - \int 10^{\log_{e} (x)} \log_{e} (10) dx \Rightarrow (1 + \log_{e} (10)) I_{1} = x 10^{\log_{e} (x)} \Rightarrow I_{1} = \frac{x . 10^{\log_{e} (x)}}{1 + \log_{e} (10)} ∴ I = {[\frac{x . 10^{\log_{e} (x)}}{1 + \log_{e} (10)}]}_{1}^{e} = [\frac{10 e - 1}{1 + \log_{e} (10)}] = \frac{10 e - 1}{\log_{e} (10 e)}$

174.

The value of $\int_{- 2}^{4} |x + 1| dx$ is equal to

175.

If $\int \frac{x + 2}{2 x^{2} + 6 x + 5} dx$ = P $\int \frac{4 x + 6}{2 x^{2} + 6 x + 5} dx$ + $\frac{1}{2} \int \frac{dx}{2 x^{2} + 6 x + 5}$ , then the values of P is

$\frac{1}{3}$
$\frac{1}{2}$
$\frac{1}{4}$
2

176.

$\int (x + 1) {(x + 2)}^{7} (x + 3) dx$ is equal to

$\frac{{(x + 2)}^{10}}{10} - \frac{{(x + 2)}^{8}}{8} + c$
$\frac{{(x + 1)}^{10}}{2} - \frac{{(x + 2)}^{8}}{8} - \frac{{(x + 3)}^{2}}{2} + c$
$\frac{{(x + 2)}^{10}}{10} + c$
$\frac{{(x + 1)}^{2}}{2} + \frac{{(x + 2)}^{8}}{8} + \frac{{(x + 3)}^{2}}{2} + c$

177.

$\int (x^{2} + 1) \sqrt{x + 1} dx$ is equal to

$\frac{{(x + 1)}^{\frac{7}{2}}}{7} - 2 \frac{{(x + 1)}^{\frac{5}{2}}}{5} + 2 \frac{{(x + 1)}^{\frac{3}{2}}}{3} + c$
$2 [\frac{{(x + 1)}^{\frac{7}{2}}}{7} - 2 \frac{{(x + 1)}^{\frac{5}{2}}}{5} + 2 \frac{{(x + 1)}^{\frac{3}{2}}}{3}] + c$
$\frac{{(x + 1)}^{\frac{7}{2}}}{7} - 2 \frac{{(x + 1)}^{\frac{5}{2}}}{5} + c$
${(x + 1)}^{\frac{7}{2}} + {(x + 1)}^{\frac{5}{2}} + {(x + 1)}^{\frac{3}{2}} + c$