What is the integral of ln(x+(1+x^2)^{1/2}) ?

The integral of ln(x+(1+x^2)^{1/2}) is xln(x+(1+x^2)^{1/2})-(1+x^2)^{1/2}+C

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integral of ln(x+(1+x^2)^{1/2})

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

x ln (x + (1 + x^{2})^{\frac{1}{2}}) - (1 + x^{2})^{\frac{1}{2}} + C

Solution steps

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

Apply Integration By Parts

: x ln (x + (1 + x^{2})^{\frac{1}{2}}) - \int \frac{x}{( 1 + x ^{2} ) ^{\frac{1}{2}}} d x

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

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

= x ln (x + (1 + x^{2})^{\frac{1}{2}}) - (1 + x^{2})^{\frac{1}{2}}

Add a constant to the solution

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

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What is the integral of ln(x+(1+x^2)^{1/2}) ?
The integral of ln(x+(1+x^2)^{1/2}) is xln(x+(1+x^2)^{1/2})-(1+x^2)^{1/2}+C