Question

∫ (x⁶+7x⁵+6x⁴+5x³+4x²+3x+1)eˣ dx=......+c ,Select correct option from the given options.
7
(a) ∑ xieˣ
i=1
6
(b) ∑ xieˣ
i=1

6
(c) ∑ ieˣ
i=0

6
(d) ∑ (xe)i
i=0

Answer 1

Answer:

option no c) is the ans for given question

Answer 2

Answer:

6

∑$x_{i}e^{x}$

i=1

Step-by-step explanation:

Let us assume that,

$\int\ {[x^{6}+7x^{5}+6x^{4}+5x^{3}+4x^{2}+3x+1]e^{x} } \, dx =A+c$......... (1) { Where c is the integration constant}.

Now, we have to find A. For this we have integrate the left hand part of the above equation.

$\int\ {[x^{6}+7x^{5}+6x^{4}+5x^{3}+4x^{2}+3x+1]e^{x} } \, dx$

= $\int\ {(x^{6}+6x^{5})e^{x} } \, dx +\int\ {(x^{5}+5x^{4})e^{x} } \, dx +\int\ {(x^{4}+4x^{3})e^{x} } \, dx +\int\ {(x^{3}+3x^{2})e^{x} } \, dx +\int\ {(x^{2}+2x^{1})e^{x} } \, dx +\int\ {(x^{1}+x^{0})e^{x} } \, dx$

= $x^{6}e^{x}+x^{5}e^{x}+x^{4}e^{x}+x^{3}e^{x}+x^{2}e^{x}+x^{1}e^{x}+c$

[Here we have applied the formula related to integration :

$\int\ {[f(x)+f'(x)]e^{x} } \, dx= e^{x}f(x)+c$ ]

= $(x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x)e^{x}+c$

                      6

Therefore, A=∑$x_{i}e^{x}$ (Answer)

                        i=1