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Solving integrals/ 1/4(x+1)^2

Integral of 1/4(x+1)^2 dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
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∫0−1(x+1)24 dx\int\limits_{0}^{-1} \frac{\left(x + 1\right)^{2}}{4}\, dx 
Integral((x + 1)^2/4, (x, 0, -1))
Detail solution

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

    ∫(x+1)24 dx=∫(x+1)2 dx4\int \frac{\left(x + 1\right)^{2}}{4}\, dx = \frac{\int \left(x + 1\right)^{2}\, dx}{4}

    1. There are multiple ways to do this integral.

      Method #1

      1. Let u=x+1u = x + 1.

        Then let du=dxdu = dx and substitute dudu:

        ∫u2 du\int u^{2}\, du

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

          ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

        Now substitute uu back in:

        (x+1)33\frac{\left(x + 1\right)^{3}}{3}

      Method #2

      1. Rewrite the integrand:

        (x+1)2=x2+2x+1\left(x + 1\right)^{2} = x^{2} + 2 x + 1

      2. Integrate term-by-term:

        1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

          ∫x2 dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

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

          ∫2x dx=2∫x dx\int 2 x\, dx = 2 \int x\, dx

          1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

          So, the result is: x2x^{2}

        1. The integral of a constant is the constant times the variable of integration:

          ∫1 dx=x\int 1\, dx = x

        The result is: x33+x2+x\frac{x^{3}}{3} + x^{2} + x

    So, the result is: (x+1)312\frac{\left(x + 1\right)^{3}}{12}

  2. Now simplify:

    (x+1)312\frac{\left(x + 1\right)^{3}}{12}

  3. Add the constant of integration:

    (x+1)312+constant\frac{\left(x + 1\right)^{3}}{12}+ \mathrm{constant}

The answer is:

(x+1)312+constant\frac{\left(x + 1\right)^{3}}{12}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                          
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 |        2                 3
 | (x + 1)           (x + 1) 
 | -------- dx = C + --------
 |    4                 12   
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/
∫(x+1)24 dx=C+(x+1)312\int \frac{\left(x + 1\right)^{2}}{4}\, dx = C + \frac{\left(x + 1\right)^{3}}{12}
Simplify
The graph
-1.00-0.90-0.80-0.70-0.60-0.50-0.40-0.30-0.20-0.100.000.5-0.5
Plot the graph f(x)
The answer [src] 
-1/12
−112- \frac{1}{12} 
=
= 
-1/12
−112- \frac{1}{12} 
-1/12
Numerical answer [src] 
-0.0833333333333333
-0.0833333333333333

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

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