Question

$\int (x^6+7x^5+6x^4+5x^3+4x^2+3x+1)e^{x}dx$ equals

• A. ${\sum }_{j=0}^6{x}^j{e}^x+c$
• B. ${\sum }_{j=1}^7{x}^j{e}^x+c$
• C. ${\sum }_{i=1}^6{x}^i{e}^x+c$
• D. ${\sum }_{i=0}^5{x}^i{e}^x+c$

Answer 1

The correct option is C ${\sum }_{i=1}^6{x}^i{e}^x+c$

$\int (x^6+7x^5+6x^4+5x^3+4x^2+3x+1)e^{x}dx$

$=\int (x^6+6x^5)e^{x}dx+\int (x^5+5x^4)e^{x}dx+\int (x^4+4x^3)e^{x}dx+\int (x^3+3x^2)e^{x}dx=\int (x^2+2x)e^{x}dx+\int (x+1)e^{x}dx$.

$\int \left[f\right(x)+f^{\prime }(x\left)\right]e^{x}dx=f\left(x\right)e^x$

Using this fact, we get

$\int (x^6+6x^5)e^{x}dx=\int (x^6+(x^6{)}^{\prime })e^{x}dx$

$=x^6\cdot e^{x}dx$

Hence G.E$=x^6{e}^x+x^5{e}^x+...\dots +x{e}^x+c$

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