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Formula
Calculate the integral
Answer
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$\int{ \left( \sqrt{ x } - \dfrac{ 1 }{ \sqrt{ x } } \right) }d{ x }$
$2 \times \dfrac { 1 } { 3 } \sqrt{ x ^ { 3 } } - 2 \sqrt{ x } + C$
Calculate the indefinite integral.
$\displaystyle\int { \sqrt{ \color{#FF6800}{ x } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { \sqrt{ x } } } } d { \color{#FF6800}{ x } }$
$ $ It is $ \int f_1(x) + f_2(x) + \cdots + f_n(x) dx = \int f_1(x) dx + \int f_2(x) dx + \cdots + \int f_n(x) dx$
$\displaystyle\int { \sqrt{ \color{#FF6800}{ x } } } d { \color{#FF6800}{ x } } \color{#FF6800}{ - } \displaystyle\int { \color{#FF6800}{ \dfrac { 1 } { \sqrt{ x } } } } d { \color{#FF6800}{ x } }$
$\displaystyle\int { \sqrt{ \color{#FF6800}{ x } } } d { \color{#FF6800}{ x } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$ $ Substitute with $ u = \sqrt{ x } $ and calculate the integral $ $
$\left [ \color{#FF6800}{ 2 } \displaystyle\int { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \sqrt{ \color{#FF6800}{ x } } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$\left [ 2 \displaystyle\int { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } } d { \color{#FF6800}{ u } } \right ] _ { u = \sqrt{ x } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$ $ Calculate the integral using the formula of $ \int{x^{n}}dx = \frac{x^{n+1}}{n+1}$
$\left [ 2 \times \color{#FF6800}{ \frac { 1 } { 2 + 1 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ] _ { u = \sqrt{ x } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$\left [ 2 \times \frac { 1 } { \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } u ^ { 2 + 1 } \right ] _ { u = \sqrt{ x } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$ $ Add $ 2 $ and $ 1$
$\left [ 2 \times \frac { 1 } { \color{#FF6800}{ 3 } } u ^ { 2 + 1 } \right ] _ { u = \sqrt{ x } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$\left [ 2 \times \frac { 1 } { 3 } u ^ { \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ] _ { u = \sqrt{ x } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$ $ Add $ 2 $ and $ 1$
$\left [ 2 \times \frac { 1 } { 3 } u ^ { \color{#FF6800}{ 3 } } \right ] _ { u = \sqrt{ x } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$\left [ \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ \frac { 1 } { 3 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 3 } } \right ] _ { \color{#FF6800}{ u } = \sqrt{ \color{#FF6800}{ x } } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$ $ Return the substituted value $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 3 } } \left ( \sqrt{ \color{#FF6800}{ x } } \right ) ^ { \color{#FF6800}{ 3 } } - \displaystyle\int { \dfrac { 1 } { \sqrt{ x } } } d { x }$
$2 \times \dfrac { 1 } { 3 } \left ( \sqrt{ x } \right ) ^ { 3 } - \displaystyle\int { \color{#FF6800}{ \dfrac { 1 } { \sqrt{ x } } } } d { \color{#FF6800}{ x } }$
$ $ Substitute with $ u = \sqrt{ x } $ and calculate the integral $ $
$2 \times \dfrac { 1 } { 3 } \left ( \sqrt{ x } \right ) ^ { 3 } - \left [ \color{#FF6800}{ 2 } \displaystyle\int { \color{#FF6800}{ 1 } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \sqrt{ \color{#FF6800}{ x } } }$
$2 \times \dfrac { 1 } { 3 } \left ( \sqrt{ x } \right ) ^ { 3 } - \left [ 2 \displaystyle\int { \color{#FF6800}{ 1 } } d { \color{#FF6800}{ u } } \right ] _ { u = \sqrt{ x } }$
$ $ The indefinite integral of $ 1 $ is $ x $ . $ $
$2 \times \dfrac { 1 } { 3 } \left ( \sqrt{ x } \right ) ^ { 3 } - \left [ 2 \color{#FF6800}{ u } \right ] _ { u = \sqrt{ x } }$
$2 \times \dfrac { 1 } { 3 } \left ( \sqrt{ x } \right ) ^ { 3 } - \left [ \color{#FF6800}{ 2 } \color{#FF6800}{ u } \right ] _ { \color{#FF6800}{ u } = \sqrt{ \color{#FF6800}{ x } } }$
$ $ Return the substituted value $ $
$2 \times \dfrac { 1 } { 3 } \left ( \sqrt{ x } \right ) ^ { 3 } - \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ x } } \right )$
$2 \times \dfrac { 1 } { 3 } \left ( \sqrt{ \color{#FF6800}{ x } } \right ) ^ { \color{#FF6800}{ 3 } } - \left ( 2 \sqrt{ x } \right )$
$ $ Calculate power $ $
$2 \times \dfrac { 1 } { 3 } \sqrt{ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 3 } } } - \left ( 2 \sqrt{ x } \right )$
$2 \times \dfrac { 1 } { 3 } \sqrt{ x ^ { 3 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ x } } \right )$
$ $ Get rid of unnecessary parentheses $ $
$2 \times \dfrac { 1 } { 3 } \sqrt{ x ^ { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ x } }$
$\color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 3 } } \sqrt{ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 3 } } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ x } }$
$ $ Add the integral constant $ C $ . $ $
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 3 } } \sqrt{ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 3 } } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ x } } \right ) \color{#FF6800}{ + } \color{#FF6800}{ C }$
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 3 } } \sqrt{ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 3 } } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ x } } \right ) \color{#FF6800}{ + } \color{#FF6800}{ C }$
$ $ Get rid of unnecessary parentheses $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 3 } } \sqrt{ \color{#FF6800}{ x } ^ { \color{#FF6800}{ 3 } } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ x } } \color{#FF6800}{ + } \color{#FF6800}{ C }$
Solution search results
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