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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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