Important Questions of Integrals Mathematics | Zigya

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Mathematics : Integrals

Multiple Choice Questions

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461.

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

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

462.

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

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

463.

$\int \frac{1 + x}{x + e^{- x}} dx$ is equal to

$\log |(x - e^{- x})|$
$\log |(x + e^{- x})|$
$\log |(1 + {xe}^{x})| + c$
${(1 + {xe}^{x})}^{2}$ + c

464.

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

${[\log (x + \sqrt{1 + x^{2}})]}^{2} + c$
$xlog (x + \sqrt{1 + x^{2}}) + c$
$\frac{1}{2} \log (x + \sqrt{1 + x^{2}}) + c$
$\frac{x}{2} \log (x + \sqrt{1 + x^{2}}) + c$

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465.

$\int \frac{dx}{\sqrt{1 - e^{2 x}}}$ is equal to

$\log |e^{- x} + \sqrt{e^{- 2 x} - 1}| + c$
$\log |e^{x} + \sqrt{e^{2 x} - 1}| + c$
$- \log |e^{- x} + \sqrt{e^{- 2 x} - 1}| + c$
$- \log |e^{- 2 x} + \sqrt{e^{- 2 x} - 1}| + c$

466.

$\int \frac{\cos (x) + xsin (x)}{x^{2} \cos (x)} dx$ is equal to

$\log |\frac{\sin (x)}{1 + \cos (x)}| + c$
$\log |\frac{\sin (x)}{x + \cos (x)}| + c$
$\log |\frac{2 \sin (x)}{x + \cos (x)}| + c$
$\log |\frac{x}{x + \cos (x)}| + c$

467.

The integral $\int_{0}^{1} \frac{2 \sin^{- 1} (\frac{x}{2})}{x} dx$ equals

$\int_{0}^{\frac{π}{6}} \frac{xdx}{\tan (x)}$
$\int_{0}^{\frac{π}{6}} \frac{2}{\tan (x)} dx$
$\int_{0}^{\frac{π}{2}} \frac{2 xdx}{\tan (x)}$
$\int_{0}^{\frac{π}{6}} \frac{2 x}{\sin (x)} dx$

468.

If $\int_{0}^{a} f (2 a - x) dx = m and \int_{0}^{a} f (x) dx = n$ , then $\int_{0}^{2 a} f (x) dx$ is equal to

2m + n
m + 2n
m - n
m + n

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469.

$\int_{- 100}^{100} f (x) dx$ is equal to

$\int_{- 100}^{100} f (x^{2}) dx$
$\int_{- 100}^{100} f (- x^{2}) dx$
$\int_{- 100}^{100} f (\frac{1}{x}) dx$
$\int_{- 100}^{100} f (- x) dx$

470.

$\int_{- 1}^{1} (e^{x^{3}} + e^{- x^{3}}) (e^{x} - e^{- x}) dx$ is equal to

$\frac{e^{2}}{2} - 2 e$
e² - 2e
2(e² - e)
0

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