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Find the integral value
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$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( 1 + x ^ {2} \right)} + C$
using substitution
$\int \arctan \left( x \right)dx$
$ $ Expand $ \arctan \left( x \right) \times 1 $ to use the partial integration formula $ $
$\int \color{#FF6800}{}\arctan \left( x \right)\color{#FF6800}{ \times 1}dx$
$\color{#FF6800}{\int \arctan \left( x \right) \times 1dx}$
$ $ To solve the integral, substitute $ v = x $ and $ u = \arctan \left( x \right) $ , and use the partial integration formula $ \int udv = uv - \int vdu $ $ $
$\color{#FF6800}{\arctan \left( x \right) \times x - \int x \times \dfrac {1} {1 + x ^ {2}}dx}$
$\arctan \left( x \right) \times x - \int \color{#FF6800}{x \times \dfrac {1} {1 + x ^ {2}}}dx$
$ $ Calculate the following $ $
$\arctan \left( x \right) \times x - \int \color{#FF6800}{\dfrac {x} {1 + x ^ {2}}}dx$
$\arctan \left( x \right) \times x - \color{#FF6800}{\int \dfrac {x} {1 + x ^ {2}}dx}$
$ $ Substitute $ t = 1 + x ^ {2} $ to simplify integral calculation $ $
$\arctan \left( x \right) \times x - \color{#FF6800}{\int \dfrac {1} {2t}dt}$
$\arctan \left( x \right) \times x - \color{#FF6800}{\int \dfrac {1} {2t}dt}$
$ $ Utilize integration $ \int a \times f(x)dx = a \times \int f(x)dx, a \in ℝ $ $ $
$\arctan \left( x \right) \times x - \color{#FF6800}{\dfrac {1} {2} \times \int \dfrac {1} {t}dt}$
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \color{#FF6800}{\int \dfrac {1} {t}dt}$
$ $ Using $ \int \dfrac {1} {x}dx = \ln{\left( |x| \right)} $ , calculate the integral $ $
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \color{#FF6800}{\ln}{\left( \color{#FF6800}{|t|} \right)}$
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( |\color{#FF6800}{t}| \right)}$
$ $ Change the substituted $ t = 1 + x ^ {2} $ again $ $
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( |\color{#FF6800}{1 + x ^ {2}}| \right)}$
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( \color{#FF6800}{|1 + x ^ {2}|} \right)}$
$ $ If the number of the absolute value sign is positive, remove the absolute value sign $ $
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( \color{#FF6800}{1 + x ^ {2}} \right)}$
$\color{#FF6800}{\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln }{\left( \color{#FF6800}{1 + x ^ {2}} \right)} \color{#FF6800}{}$
$ $ Add integral constant $ C \in ℝ $ $ $
$\color{#FF6800}{\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln }{\left( \color{#FF6800}{1 + x ^ {2}} \right)} \color{#FF6800}{ + C}$
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( 1 + x ^ {2} \right)} + C$
Using partial integration
$\int \arctan \left( x \right)dx$
$ $ Expand $ \arctan \left( x \right) \times 1 $ to use the partial integration formula $ $
$\int \color{#FF6800}{}\arctan \left( x \right)\color{#FF6800}{ \times 1}dx$
$\color{#FF6800}{\int \arctan \left( x \right) \times 1dx}$
$ $ To solve the integral, substitute $ v = x $ and $ u = \arctan \left( x \right) $ , and use the partial integration formula $ \int udv = uv - \int vdu $ $ $
$\color{#FF6800}{\arctan \left( x \right) \times x - \int x \times \dfrac {1} {1 + x ^ {2}}dx}$
$\arctan \left( x \right) \times x - \int \color{#FF6800}{x \times \dfrac {1} {1 + x ^ {2}}}dx$
$ $ Calculate the following $ $
$\arctan \left( x \right) \times x - \int \color{#FF6800}{\dfrac {x} {1 + x ^ {2}}}dx$
$\arctan \left( x \right) \times x - \color{#FF6800}{\int \dfrac {x} {1 + x ^ {2}}dx}$
$ $ Substitute $ t = 1 + x ^ {2} $ to simplify integral calculation $ $
$\arctan \left( x \right) \times x - \color{#FF6800}{\int \dfrac {1} {2t}dt}$
$\arctan \left( x \right) \times x - \color{#FF6800}{\int \dfrac {1} {2t}dt}$
$ $ Utilize integration $ \int a \times f(x)dx = a \times \int f(x)dx, a \in ℝ $ $ $
$\arctan \left( x \right) \times x - \color{#FF6800}{\dfrac {1} {2} \times \int \dfrac {1} {t}dt}$
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \color{#FF6800}{\int \dfrac {1} {t}dt}$
$ $ Using $ \int \dfrac {1} {x}dx = \ln{\left( |x| \right)} $ , calculate the integral $ $
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \color{#FF6800}{\ln}{\left( \color{#FF6800}{|t|} \right)}$
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( |\color{#FF6800}{t}| \right)}$
$ $ Change the substituted $ t = 1 + x ^ {2} $ again $ $
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( |\color{#FF6800}{1 + x ^ {2}}| \right)}$
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( \color{#FF6800}{|1 + x ^ {2}|} \right)}$
$ $ If the number of the absolute value sign is positive, remove the absolute value sign $ $
$\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln{\left( \color{#FF6800}{1 + x ^ {2}} \right)}$
$\color{#FF6800}{\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln }{\left( \color{#FF6800}{1 + x ^ {2}} \right)} \color{#FF6800}{}$
$ $ Add integral constant $ C \in ℝ $ $ $
$\color{#FF6800}{\arctan \left( x \right) \times x - \dfrac {1} {2} \times \ln }{\left( \color{#FF6800}{1 + x ^ {2}} \right)} \color{#FF6800}{ + C}$
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