Mathematics

# $\int \left( x ^ { 6 } + 7 x ^ { 5 } + 6 x ^ { 4 } + 5 x ^ { 3 } + 4 x ^ { 2 } + 3 x + 1 \right) e ^ { x } d x$ equals

$\sum _ { i = 1 } ^ { 6 } x ^ { i } e ^ { x } + c$

##### SOLUTION
$\displaystyle\int (x^6+7x^5+6x^4+5x^3+4x^2+3x+1)e^xdx$
$=\displaystyle\int (x^6+6x^5)e^xdx+\displaystyle\int (x^5+5x^4)e^xdx+\displaystyle\int (x^4+4x^3)e^xdx+\displaystyle\int (x^3+3x^2)e^xdx=\displaystyle\int (x^2+2x)e^xdx+\displaystyle\int (x+1)e^xdx$.
$\displaystyle\int [f(x)+f'(x)]e^xdx=f(x)e^x$
Using this fact, we get
$\displaystyle\int (x^6+6x^5)e^xdx=\displaystyle\int (x^6+(x^6)')e^xdx$
$=x^6\cdot e^xdx$
Hence G.E$=x^6e^x+x^5e^x+...…+xe^x+c$
$=\displaystyle\sum^6_{i=1}x^ie^x+c$.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium
$I=\int ({1+x+x^{2}})dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int e^{-x}\mbox{cosec}\ x\ (\cot x+1)dx=$
• A. $e^{x}\mbox{cosec}\ x+c$
• B. $e^{x}\cot x+c$
• C. $-e^{-x}\cot x+c$
• D. $-e^{-x}\mbox{cosec}\ x+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate the function    $\displaystyle \frac {xe^x}{(1+x)^2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle \lim_{n \rightarrow \infty} \frac {(1^{2}+2^{2}+3^{2}+...+n^{2})(1^{3}+2^{3}+3^{3}+...+n^{3})}{(1^{6}+2^{6}+3^{6}+...+n^{6})}=?$
• A. $\displaystyle \frac {1}{6}$
• B. $\displaystyle \frac {1}{12}$
• C. $\displaystyle \frac {1}{7}$
• D. $\displaystyle \frac {7}{12}$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$