how to integrate (1+4x^2)^(1/2)?

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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We know that:

`int sqrt(a^2+-x^2) dx = 1/2 xsqrt(a^2+-x^2)+-1/2 a^2 ln(x+sqrt(a^2+-x^2))`

`==> int (1+4x^2)^(1/2) dx = int sqrt(1+4x^2) dx=int sqrt(4(1+4x^2)/4) dx`

`==> 2 int sqrt(1/4 + x^2) dx`

`==> a = 1/2`

`==>2 int sqrt(1/4 + x^2)= 2[1/2 xsqrt(1/4+x^2) + (1/2)(1/4) ln(x+sqrt(1/4 + x^2))]`

`==> 2int (1/4+x^2) dx = xsqrt(1/4 + x^2) + (1/4)ln(x+sqrt(1/4+x^2))`

`==> int (1+4x^2)^(1/2) dx = xsqt(1/4 +x^2) + 1/4 ln (x+sqrt(1/4 + x^2))`

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