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Calculus · Definite-Integrals

Integrate sqrt(x^2 + 1) from 0 to 1

\int_{0}^{1} \sqrt{x^{2} + 1}\, dx

Hatua kwa hatua

  1. \int_{0}^{1} \sqrt{x^{2} + 1}\, dx

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int \sqrt{x^{2} + 1}\, dx = \frac{x \sqrt{x^{2} + 1}}{2} + \frac{\operatorname{asinh}{\left(x \right)}}{2}

    SqrtQuadratic rule.

  3. F(1) - F(0) = \left(\frac{\log{\left(1 + \sqrt{2} \right)}}{2} + \frac{\sqrt{2}}{2}\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

  4. = \frac{\log{\left(1 + \sqrt{2} \right)}}{2} + \frac{\sqrt{2}}{2} \approx 1.1478

    Simplify.

Lafunua jibu
\frac{\log{\left(1 + \sqrt{2} \right)}}{2} + \frac{\sqrt{2}}{2}