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Solving integrals/ 1/((1+x)*sqrt(x+1))

Integral of 1/((1+x)*sqrt(x+1)) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫011x+1(x+1) dx\int\limits_{0}^{1} \frac{1}{\sqrt{x + 1} \left(x + 1\right)}\, dx 
Integral(1/((1 + x)*sqrt(x + 1)), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      1x+1(x+1)=1xx+1+x+1\frac{1}{\sqrt{x + 1} \left(x + 1\right)} = \frac{1}{x \sqrt{x + 1} + \sqrt{x + 1}}

    2. Let u=x+1u = \sqrt{x + 1}.

      Then let du=dx2x+1du = \frac{dx}{2 \sqrt{x + 1}} and substitute 2du2 du:

      ∫2u2 du\int \frac{2}{u^{2}}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫1u2 du=2∫1u2 du\int \frac{1}{u^{2}}\, du = 2 \int \frac{1}{u^{2}}\, du

        1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫1u2 du=−1u\int \frac{1}{u^{2}}\, du = - \frac{1}{u}

        So, the result is: −2u- \frac{2}{u}

      Now substitute uu back in:

      −2x+1- \frac{2}{\sqrt{x + 1}}

    Method #2

    1. Rewrite the integrand:

      1x+1(x+1)=1xx+1+x+1\frac{1}{\sqrt{x + 1} \left(x + 1\right)} = \frac{1}{x \sqrt{x + 1} + \sqrt{x + 1}}

    2. Let u=x+1u = \sqrt{x + 1}.

      Then let du=dx2x+1du = \frac{dx}{2 \sqrt{x + 1}} and substitute 2du2 du:

      ∫2u2 du\int \frac{2}{u^{2}}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫1u2 du=2∫1u2 du\int \frac{1}{u^{2}}\, du = 2 \int \frac{1}{u^{2}}\, du

        1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫1u2 du=−1u\int \frac{1}{u^{2}}\, du = - \frac{1}{u}

        So, the result is: −2u- \frac{2}{u}

      Now substitute uu back in:

      −2x+1- \frac{2}{\sqrt{x + 1}}

  2. Add the constant of integration:

    −2x+1+constant- \frac{2}{\sqrt{x + 1}}+ \mathrm{constant}

The answer is:

−2x+1+constant- \frac{2}{\sqrt{x + 1}}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫1x+1(x+1) dx=C−2x+1\int \frac{1}{\sqrt{x + 1} \left(x + 1\right)}\, dx = C - \frac{2}{\sqrt{x + 1}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
      ___
2 - \/ 2
2−22 - \sqrt{2} 
=
= 
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2 - \/ 2
2−22 - \sqrt{2} 
2 - sqrt(2)
Numerical answer [src] 
0.585786437626905
0.585786437626905

    Use the examples entering the upper and lower limits of integration.

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