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Indefinite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration3 Marks
Question
Evaluate : $\int x^2 \cdot 5^x d x$
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Answer

$
\begin{aligned}
\text {I} & =x^2 \cdot \int 5^x \cdot d x-\int \frac{d}{d x} \cdot x^2 \cdot \int 5^x \cdot d x \cdot d x \\
& =x^2 \cdot 5^x \cdot \frac{1}{\log 5}-\int 2 x \cdot 5^x \cdot \frac{1}{\log 5} \cdot d x \\
& =\frac{1}{\log 5} \cdot x^2 \cdot 5^x-\frac{2}{\log 5}\left\{x \cdot \int 5^x \cdot d x-\int \frac{d}{d x} \cdot x \cdot \int 5^x \cdot d x \cdot d x\right\} \\
& =\frac{1}{\log 5} \cdot x^2 \cdot 5^x-\frac{2}{\log 5}\left\{x \cdot 5^x \cdot \frac{1}{\log 5}-\int(1)\left(5^x \cdot \frac{1}{\log 5}\right) \cdot d x\right\} \\
& =\frac{1}{\log 5} \cdot x^2 \cdot 5^x-\frac{2}{\log 5}\left\{\frac{1}{\log 5} \cdot x \cdot 5^x \cdot \int \frac{1}{\log 5} \cdot 5^x \cdot d x\right\} \\
& =\frac{1}{\log 5} \cdot x^2 \cdot 5^x-\frac{2}{\log 5}\left\{\frac{1}{\log 5} \cdot x \cdot 5^x \cdot-\frac{1}{\log 5} \cdot 5^x \cdot \frac{1}{\log 5}\right\}+c \\
& =\frac{1}{\log 5} \cdot x^2 \cdot 5^x-\frac{2}{(\log 5)^2} \cdot x \cdot 5^x+\frac{2}{(\log 5)^3} \cdot 5^x+c \\
& =\frac{5^x}{\log 5} \cdot\left\{x^2-\frac{2 x}{\log 5}+\frac{2}{(\log 5)^2}\right\}+c
\end{aligned}
$

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