What is the integral of x+sqrt(1+x^2) ?

The integral of x+sqrt(1+x^2) is (x^2)/2+1/2 xsqrt(1+x^2)+1/2 ln|x+sqrt(1+x^2)|+C

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integral of x+sqrt(1+x^2)

\int x + 1 + x^{2} d x

\frac{x ^{2}}{2} + \frac{1}{2} x 1 + x^{2} + \frac{1}{2} ln x + 1 + x^{2} + C

Solution steps

\int x + 1 + x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x d x + \int 1 + x^{2} d x

\int x d x = \frac{x ^{2}}{2}

\int 1 + x^{2} d x = \frac{1}{2} x 1 + x^{2} + \frac{1}{2} ln x + 1 + x^{2}

= \frac{x ^{2}}{2} + \frac{1}{2} x 1 + x^{2} + \frac{1}{2} ln x + 1 + x^{2}

Add a constant to the solution

= \frac{x ^{2}}{2} + \frac{1}{2} x 1 + x^{2} + \frac{1}{2} ln x + 1 + x^{2} + C

What is the integral of x+sqrt(1+x^2) ?
The integral of x+sqrt(1+x^2) is (x^2)/2+1/2 xsqrt(1+x^2)+1/2 ln|x+sqrt(1+x^2)|+C