KCET · Maths · Indefinite Integration
The value of \(\int e^{x}\left(x^{5}+5 x^{4}+1\right) \cdot d x\) is
- A \(e^{x} \cdot x^{5}+c\)
- B \(e^{x} \cdot x^{5}+e^{x}+c\)
- C \(e^{x+1} \cdot x^{5}+c\)
- D \(5 x^{4} \cdot e^{x}+c\)
Answer & Solution
Correct Answer
(B) \(e^{x} \cdot x^{5}+e^{x}+c\)
Step-by-step Solution
Detailed explanation
Let \(I=\int e^{x}\left(x^{5}+5 x^{4}+1\right) d x\)
\[
\begin{aligned}
&=\int e^{x} x^{5} d x+5 \int e^{x} x^{4} d x+\int e^{x} d x \\
&=x^{5} e^{x}-\int 5 x^{4} e^{x} d x+5 \int e^{x} x^{4} d x+e^{x} \\
&=x^{5} e^{x}+e^{x}+c
\end{aligned}
\]
\[
\begin{aligned}
&=\int e^{x} x^{5} d x+5 \int e^{x} x^{4} d x+\int e^{x} d x \\
&=x^{5} e^{x}-\int 5 x^{4} e^{x} d x+5 \int e^{x} x^{4} d x+e^{x} \\
&=x^{5} e^{x}+e^{x}+c
\end{aligned}
\]
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