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प्रश्न
Evaluate the following : `int_0^4 [sqrt(x^2 + 2x + 3]]^-1*dx`
उत्तर
Let I = `int_0^4 [sqrt(x^2 + 2x + 3]]^-1*dx`
= `int_0^4 1/sqrt(x^2 + 2x + 1 + 2)*dx`
= `int_0^4 1/sqrt((x + 1)^2 + 2)*dx`
= `[log [x + 1 + sqrt((x + 1)^2 + 2)]_0^4`
= `log[4 + 1 + sqrt(5^2 + 2)] - log[0 + 1 + sqrt(1^2 + 2)]`
= `log(5 + 3sqrt(3)) - log(1 + sqrt(3))`
= `log((5 + 3sqrt(3))/(1 + sqrt(3)))`.
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