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Formula
Calculate the integral
Answer
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$\int{ \sqrt[3]{ t } \left( 5t ^{ 2 } -3t+2 \right) }d{ t }$
$15 \times \dfrac { 1 } { 10 } \sqrt[ 3 ]{ t ^ { 10 } } - 9 \times \dfrac { 1 } { 7 } \sqrt[ 3 ]{ t ^ { 7 } } + 6 \times \dfrac { 1 } { 4 } \sqrt[ 3 ]{ t ^ { 4 } }$
Calculate the integral
$\displaystyle\int { \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } \left ( \color{#FF6800}{ 5 } \color{#FF6800}{ t } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ t } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) } d { \color{#FF6800}{ t } }$
$ $ Substitute with $ u = \sqrt[ 3 ]{ t } $ and calculate the integral $ $
$\left [ \displaystyle\int { \color{#FF6800}{ 15 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 9 } } \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } } \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 3 } } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } }$
$\left [ \displaystyle\int { \color{#FF6800}{ 15 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 9 } } \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } } \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 3 } } } d { \color{#FF6800}{ u } } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ 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$
$\left [ \displaystyle\int { \color{#FF6800}{ 15 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 9 } } } d { \color{#FF6800}{ u } } \color{#FF6800}{ + } \displaystyle\int { \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } } } d { \color{#FF6800}{ u } } \color{#FF6800}{ + } \displaystyle\int { \color{#FF6800}{ 6 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 3 } } } d { \color{#FF6800}{ u } } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ \displaystyle\int { \color{#FF6800}{ 15 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 9 } } } d { \color{#FF6800}{ u } } + \displaystyle\int { - 9 u ^ { 6 } } d { u } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ It is $ \int c f(x) dx = c \int f(x) dx$
$\left [ \color{#FF6800}{ 15 } \displaystyle\int { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 9 } } } d { \color{#FF6800}{ u } } + \displaystyle\int { - 9 u ^ { 6 } } d { u } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \displaystyle\int { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 9 } } } d { \color{#FF6800}{ u } } + \displaystyle\int { - 9 u ^ { 6 } } d { u } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Calculate the integral using the formula of $ \int{x^{n}}dx = \frac{x^{n+1}}{n+1}$
$\left [ 15 \times \color{#FF6800}{ \frac { 1 } { 9 + 1 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 9 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } + \displaystyle\int { - 9 u ^ { 6 } } d { u } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { \color{#FF6800}{ 9 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } u ^ { 9 + 1 } + \displaystyle\int { - 9 u ^ { 6 } } d { u } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Add $ 9 $ and $ 1$
$\left [ 15 \times \frac { 1 } { \color{#FF6800}{ 10 } } u ^ { 9 + 1 } + \displaystyle\int { - 9 u ^ { 6 } } d { u } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { \color{#FF6800}{ 9 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } + \displaystyle\int { - 9 u ^ { 6 } } d { u } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Add $ 9 $ and $ 1$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { \color{#FF6800}{ 10 } } + \displaystyle\int { - 9 u ^ { 6 } } d { u } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } + \displaystyle\int { \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } } } d { \color{#FF6800}{ u } } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ 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$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } \color{#FF6800}{ - } \displaystyle\int { \color{#FF6800}{ 9 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } } } d { \color{#FF6800}{ u } } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \displaystyle\int { \color{#FF6800}{ 9 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } } } d { \color{#FF6800}{ u } } + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ It is $ \int c f(x) dx = c \int f(x) dx$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( \color{#FF6800}{ 9 } \displaystyle\int { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } } } d { \color{#FF6800}{ u } } \right ) + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \displaystyle\int { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } } } d { \color{#FF6800}{ u } } \right ) + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Calculate the integral using the formula of $ \int{x^{n}}dx = \frac{x^{n+1}}{n+1}$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \color{#FF6800}{ \frac { 1 } { 6 + 1 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 6 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ) + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { \color{#FF6800}{ 6 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } u ^ { 6 + 1 } \right ) + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Add $ 6 $ and $ 1$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { \color{#FF6800}{ 7 } } u ^ { 6 + 1 } \right ) + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { \color{#FF6800}{ 6 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ) + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Add $ 6 $ and $ 1$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { \color{#FF6800}{ 7 } } \right ) + \displaystyle\int { 6 u ^ { 3 } } d { u } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { 7 } \right ) + \displaystyle\int { \color{#FF6800}{ 6 } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 3 } } } d { \color{#FF6800}{ u } } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ It is $ \int c f(x) dx = c \int f(x) dx$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { 7 } \right ) + \color{#FF6800}{ 6 } \displaystyle\int { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 3 } } } d { \color{#FF6800}{ u } } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { 7 } \right ) + 6 \displaystyle\int { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 3 } } } d { \color{#FF6800}{ u } } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Calculate the integral using the formula of $ \int{x^{n}}dx = \frac{x^{n+1}}{n+1}$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { 7 } \right ) + 6 \times \color{#FF6800}{ \frac { 1 } { 3 + 1 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { 7 } \right ) + 6 \times \frac { 1 } { \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } u ^ { 3 + 1 } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Add $ 3 $ and $ 1$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { 7 } \right ) + 6 \times \frac { 1 } { \color{#FF6800}{ 4 } } u ^ { 3 + 1 } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { 7 } \right ) + 6 \times \frac { 1 } { 4 } u ^ { \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$ $ Add $ 3 $ and $ 1$
$\left [ 15 \times \frac { 1 } { 10 } u ^ { 10 } - \left ( 9 \times \frac { 1 } { 7 } u ^ { 7 } \right ) + 6 \times \frac { 1 } { 4 } u ^ { \color{#FF6800}{ 4 } } \right ] _ { u = \sqrt[ 3 ]{ t } }$
$\left [ \color{#FF6800}{ 15 } \color{#FF6800}{ \times } \color{#FF6800}{ \frac { 1 } { 10 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ \frac { 1 } { 7 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 7 } } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ \times } \color{#FF6800}{ \frac { 1 } { 4 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 4 } } \right ] _ { \color{#FF6800}{ u } = \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } }$
$ $ Return the substituted value $ $
$\color{#FF6800}{ 15 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 10 } } \left ( \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } \right ) ^ { \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 7 } } \left ( \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } \right ) ^ { \color{#FF6800}{ 7 } } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 4 } } \left ( \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } \right ) ^ { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ 15 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 10 } } \left ( \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } \right ) ^ { \color{#FF6800}{ 10 } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 7 } } \left ( \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } \right ) ^ { \color{#FF6800}{ 7 } } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 4 } } \left ( \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } } \right ) ^ { \color{#FF6800}{ 4 } }$
$ $ Simplify the expression $ $
$\color{#FF6800}{ 15 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 10 } } \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } ^ { \color{#FF6800}{ 10 } } } \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 7 } } \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } ^ { \color{#FF6800}{ 7 } } } \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { 4 } } \sqrt[ \color{#FF6800}{ 3 } ]{ \color{#FF6800}{ t } ^ { \color{#FF6800}{ 4 } } }$
Solution search results
search-thumbnail-If the sum of two consecutive 
numbers is $45$ and one number is $X$ 
.This statement in the form of 
equation $1s:$ 
$\left(1$ Point) $\right)$ 
$○5x+1$ $1eft\left(x+1$ $r1gnt\right)=45s$ 
$○sx+1ef\left(x+2$ $r1gnt\right)=145s$ 
$sx+1x=45s$
7th-9th grade
Algebra
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