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Solving integrals/ 1/sqrt(1-x^2)

Integral of 1/sqrt(1-x^2) dx

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The solution

You have entered [src] 
  1             
  /             
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 |          2   
 |  /     2\    
 |  \1 - x /    
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0
∫01(1−x2)2t dx\int\limits_{0}^{1} \frac{\left(1 - x^{2}\right)^{2}}{t}\, dx 
Integral((1 - x^2)^2/t, (x, 0, 1))
Detail solution

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

    ∫(1−x2)2t dx=∫(1−x2)2 dxt\int \frac{\left(1 - x^{2}\right)^{2}}{t}\, dx = \frac{\int \left(1 - x^{2}\right)^{2}\, dx}{t}

    1. Rewrite the integrand:

      (1−x2)2=x4−2x2+1\left(1 - x^{2}\right)^{2} = x^{4} - 2 x^{2} + 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:

        ∫x4 dx=x55\int x^{4}\, dx = \frac{x^{5}}{5}

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

        ∫(−2x2) dx=−2∫x2 dx\int \left(- 2 x^{2}\right)\, dx = - 2 \int x^{2}\, dx

        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}

        So, the result is: −2x33- \frac{2 x^{3}}{3}

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

        ∫1 dx=x\int 1\, dx = x

      The result is: x55−2x33+x\frac{x^{5}}{5} - \frac{2 x^{3}}{3} + x

    So, the result is: x55−2x33+xt\frac{\frac{x^{5}}{5} - \frac{2 x^{3}}{3} + x}{t}

  2. Now simplify:

    x(3x4−10x2+15)15t\frac{x \left(3 x^{4} - 10 x^{2} + 15\right)}{15 t}

  3. Add the constant of integration:

    x(3x4−10x2+15)15t+constant\frac{x \left(3 x^{4} - 10 x^{2} + 15\right)}{15 t}+ \mathrm{constant}

The answer is:

x(3x4−10x2+15)15t+constant\frac{x \left(3 x^{4} - 10 x^{2} + 15\right)}{15 t}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                           3    5
 |         2              2*x    x 
 | /     2\           x - ---- + --
 | \1 - x /                3     5 
 | --------- dx = C + -------------
 |     t                    t      
 |                                 
/
∫(1−x2)2t dx=C+x55−2x33+xt\int \frac{\left(1 - x^{2}\right)^{2}}{t}\, dx = C + \frac{\frac{x^{5}}{5} - \frac{2 x^{3}}{3} + x}{t}
Simplify
The answer [src] 
 8  
----
15*t
815t\frac{8}{15 t} 
=
= 
 8  
----
15*t
815t\frac{8}{15 t} 
8/(15*t)

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

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