A) \[{{e}^{x}}[{{x}^{5}}+5{{x}^{4}}+20{{x}^{3}}+60{{x}^{2}}+120x+120]+C\]
B) \[{{e}^{x}}[{{x}^{5}}-5{{x}^{4}}-20{{x}^{3}}-60{{x}^{2}}-120x-120]+C\]
C) \[{{e}^{x}}[{{x}^{5}}-5{{x}^{4}}+20{{x}^{3}}-60{{x}^{2}}+120x-120]+C\]
D) \[{{e}^{x}}[{{x}^{5}}-5{{x}^{4}}+20{{x}^{3}}-60{{x}^{2}}-120x+120]+C\]
Correct Answer: C
Solution :
Let \[I=\int{{{e}^{x}}\,.{{x}^{5}}\,\,dx}\] \[={{e}^{x}}.{{x}^{5}}-\int{{{e}^{x}}.5{{x}^{4}}dx}\] \[={{e}^{x}}.{{x}^{5}}-5[{{e}^{x}}.{{x}^{4}}-\int{{{e}^{x}}.5{{x}^{4}}dx}]\] \[={{e}^{x}}.{{x}^{5}}-5{{e}^{x}}.{{x}^{4}}+20[{{e}^{x}}.{{x}^{3}}-\int{{{e}^{x}}.5{{x}^{4}}dx}]\] \[={{e}^{x}}.{{x}^{5}}-5{{e}^{x}}.{{x}^{4}}+20{{e}^{x}}.{{x}^{3}}-60[{{e}^{x}}.{{x}^{2}}-\] \[\int{{{e}^{x}}.2x\,dx}]\] \[={{e}^{x}}.{{x}^{5}}-5{{e}^{x}}.{{x}^{4}}+20{{e}^{x}}.{{x}^{3}}-60{{e}^{x}}.{{x}^{2}}\] \[+20[{{e}^{x}}.x-{{e}^{x}}]+C\] \[={{e}^{x}}[{{x}^{5}}-5{{x}^{4}}+20{{x}^{3}}-60{{x}^{2}}+\] \[120x-120]+C\]
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