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Mathematics
Definite integration by methods of indefinite integration
How do you find the definite integral $\int\limits_{\dfrac{\pi }{2}}^{\dfrac{5\pi }{2}}{{{x}^{2}}\cos \left( \dfrac{1}{5}x \right)}dx$?

Mathematics
Definite integration by methods of indefinite integration
How to Integrate $\dfrac{7}{2}$ & $\dfrac{1}{2}$ of $\dfrac{1}{{{\left( 5+4x-{{x}^{2}} \right)}^{\dfrac{3}{2}}}}dx$?

Mathematics
Definite integration by methods of indefinite integration
How do you evaluate the definite integral $\int{\left| {{x}^{2}}-4x+3 \right|dx}$ from $\left[ 0,4 \right]$?

Mathematics
Definite integration by methods of indefinite integration
Solve the following definite integral:
$\int\limits_0^{\dfrac{1}{2}} {\dfrac{{dx}}{{(1 + {x^2})\sqrt {1 - {x^2}} }}}$
A) $\dfrac{1}{{\sqrt 2 }}{\tan ^{ - 1}}\sqrt {\dfrac{2}{3}}$
B)$\dfrac{2}{{\sqrt 2 }}{\tan ^{ - 1}}(\dfrac{3}{{\sqrt 2 }})$
C) $\dfrac{{\sqrt 2 }}{2}{\tan ^{ - 1}}(\dfrac{3}{2})$
D) $\dfrac{{\sqrt 2 }}{2}{\tan ^{ - 1}}(\dfrac{{\sqrt 3 }}{2})$

Mathematics
Definite integration by methods of indefinite integration
Evaluate the following definite integral:
$\int\limits_{0}^{1}{\dfrac{x}{{{x}^{2}}+1}}dx$

Mathematics
Definite integration by methods of indefinite integration
The integral $\int\limits_{\dfrac{\pi }{4}}^{\dfrac{\pi }{6}} {\dfrac{{dx}}{{\sin 2x\left( {{{\tan }^5}x + {{\cot }^5}x} \right)}}}$ equals :
A.$\dfrac{1}{{10}}\left( {\dfrac{\pi }{4} - {{\tan }^{ - 1}}\left( {\dfrac{1}{{9\sqrt 3 }}} \right)} \right)$
B.$\dfrac{1}{{10}}\left( {\dfrac{\pi }{4} - {{\tan }^{ - 1}}\left( {\dfrac{1}{{3\sqrt 3 }}} \right)} \right)$
C.$\dfrac{\pi }{{10}}$
D.$\dfrac{1}{{20}}{\tan ^{ - 1}}\left( {\dfrac{1}{{9\sqrt 3 }}} \right)$

Mathematics
Definite integration by methods of indefinite integration
Evaluate the definite integral given as $\int\limits_{2}^{5}{\left[ \left| x-2 \right|+\left| x-3 \right|+\left| x-5 \right| \right]}dx$.
Mathematics
Definite integration by methods of indefinite integration
Find the value of integration of the given function: $\int\limits_{ - \dfrac{\pi }{4}}^{\dfrac{\pi }{4}} {\dfrac{{dx}}{{1 + \cos 2x}} = }$
a) 1.
b) 2.
c) 3.
d) 4.

Mathematics
Definite integration by methods of indefinite integration
If $f\left( x \right) = x - \left[ x \right]$, for every real number x, where $\left[ x \right]$ is integral part of x. then
$\int\limits_{ - 1}^1 {f\left( x \right)dx}$ is

${\text{A}}{\text{. 0}} \\ {\text{B}}{\text{. 1}} \\ {\text{C}}{\text{. 2 }} \\ {\text{D}}{\text{. 3}} \\$

Mathematics
Definite integration by methods of indefinite integration
Evaluate the following definite integrals:
$\int\limits_0^1 {\dfrac{{1 - x}}{{1 + x}}dx}$
Mathematics
Definite integration by methods of indefinite integration
Evaluate the value of the integral $\int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx$.
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