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प्रश्न
Solve the following : `int_0^4 (1)/sqrt(x^2 + 2x + 3)*dx`
उत्तर
Let I = `int_0^4 (1)/sqrt(x^2 + 2x + 3)*dx`
= `int_0^4 (1)/sqrt(x^2 + 2x + 1 - 1 + 3)*dx`
= `int_0^4 (1)/((sqrt(x + 1))^2 + 2)*dx`
= `int_0^4 (1)/sqrt((x + 1)^2 + (sqrt(2))^2)*dx`
= `[log |x + 1 + sqrt((x + 1)62 + (sqrt(2))^2|]_0^4`
= `log |5 + sqrt(27)| - log| 1 + sqrt(3)|`
= `log |5 + 3sqrt(3)| - log| 1 + sqrt(3)|`
∴ I = `log |(5 + 3sqrt(3))/(1 + sqrt(3))|`.
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