If I m = ∫ 1 e ( ln x ) m dx , where m ∈ N , then I 10 + 10 I 9 is equal to -

I 10 = ∫ 1 e 1 ⋅ ( ln x ) 10 dx = [ ( ln x ) 10 x ] 1 e − ∫ 1 e 10 ( ln x ) 9 ⋅ x 1 ⋅ xdx = e − 0 − 10 ∫ 1 e ( ln x ) 9 dx = e − 10 I 9 + 10 I 9 = e

Mathematics - Area Under Curves Question with Solution | TestHub

MathematicsArea Under CurvesArea of Curve along axisMedium2 minPYQ_2010

MathematicsMediumsingle choice

If $I_{m} = \int_{1}^{e} (ln x)^{m} dx$ , where $m \in N$ , then $I_{10} + 10 I_{9}$ is equal to -

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Answer:

Solution:

$I_{10} = \int_{1}^{e} 1 \cdot (ln x)^{10} dx = [(ln x)^{10} x]_{1}^{e} - \int_{1}^{e} 10 (ln x)^{9} \cdot \frac{1}{x} \cdot xdx = e - 0 - 10 \int_{1}^{e} (ln x)^{9} dx = e - 10 I_{9} + 10 I_{9} = e$

Stream:BITSATSubject:MathematicsTopic:Area Under CurvesSubtopic:Area of Curve along axis

⏱ 2mℹ️ Source: PYQ_2010

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