Question

Evaluate integration 1/2x^2+3 ×dx​

Answer 1

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$\int \: \frac{1}{2 {x}^{2} + 3} dx \\ = \frac{1}{2} \int \: \frac{1}{ {x}^{2} + \frac{3}{2} } \\ = \frac{1}{2} \int \: \frac{1}{ {x}^{2} + ( { \frac{ \sqrt{3} }{ \sqrt{2} } })^{2} } \\ = \frac{1}{2} . \frac{1}{ \frac{ \sqrt{3} }{ \sqrt{2} } } {tan}^{ - 1} \frac{x}{ \frac{ \sqrt{3} }{ \sqrt{2} } } \\ = \frac{ \sqrt{2} }{2 \sqrt{3} } {tan}^{ - 1} \frac{ \sqrt{2} x}{ \sqrt{3} } \\ = \frac{1}{ \sqrt{6} } {tan}^{ - 1} \frac{ \sqrt{2} x}{ \sqrt{3} } + c$

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