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1 ∫ 0 2 X + 3 5 X 2 + 1 D X

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Question

\[\int\limits_0^1 \frac{2x + 3}{5 x^2 + 1} dx\]

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Solution

\[Let\ I = \int_0^1 \frac{2x + 3}{5 x^2 + 1}\ d x . Then, \]
\[I = \int_0^1 \frac{2x}{5 x^2 + 1} d x + \int_0^1 \frac{3}{5 x^2 + 1} d x\]
\[ \Rightarrow I = \frac{1}{5} \int_0^1 \frac{10x}{5 x^2 + 1} d x + 3 \int_0^1 \frac{1}{\left( \sqrt{5}x \right)^2 + 1^2} d x\]
\[ \Rightarrow I = \frac{1}{5} \left[ \log \left( 5 x^2 + 1 \right) \right]_0^1 + \frac{3}{\sqrt{5}} \left[ \tan^{- 1} \left( \sqrt{5}x \right) \right]_0^1 \]
\[ \Rightarrow I = \frac{1}{5} \log 6 + \frac{3}{\sqrt{5}} \tan^{- 1} \sqrt{5}\]

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Definite Integrals

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Chapter 19: Definite Integrals - Exercise 20.1 [Page 17]

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R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12

Chapter 19 Definite Integrals
Exercise 20.1 | Q 38 | Page 17

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