What is the integral of log_{10}(a^2+x^2) ?

The integral of log_{10}(a^2+x^2) is xlog_{10}(a^2+x^2)+2/(ln(10))aarctan(x/a)-(2x)/(ln(10))+C

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integral of log_{10}(a^2+x^2)

integral

lo g_{10} (a^{2} + x^{2})

x lo g_{10} (a^{2} + x^{2}) + \frac{2}{ln ( 10 )} a arctan (\frac{x}{a}) - \frac{2 x}{ln ( 10 )} + C

Solution steps

\int lo g_{10} (a^{2} + x^{2}) d x

Apply Integration By Parts

: x lo g_{10} (a^{2} + x^{2}) - \int \frac{2 x ^{2}}{ln ( 10 ) ( a ^{2} + x ^{2} )} d x

= x lo g_{10} (a^{2} + x^{2}) - \int \frac{2 x ^{2}}{ln ( 10 ) ( a ^{2} + x ^{2} )} d x

\int \frac{2 x ^{2}}{ln ( 10 ) ( a ^{2} + x ^{2} )} d x = - \frac{2}{ln ( 10 )} a arctan (\frac{x}{a}) + \frac{2 x}{ln ( 10 )}

= x lo g_{10} (a^{2} + x^{2}) - (- \frac{2}{ln ( 10 )} a arctan (\frac{x}{a}) + \frac{2 x}{ln ( 10 )})

Simplify

= x lo g_{10} (a^{2} + x^{2}) + \frac{2}{ln ( 10 )} a arctan (\frac{x}{a}) - \frac{2 x}{ln ( 10 )}

Add a constant to the solution

= x lo g_{10} (a^{2} + x^{2}) + \frac{2}{ln ( 10 )} a arctan (\frac{x}{a}) - \frac{2 x}{ln ( 10 )} + C

\frac{x ^{3}}{( x ^{2} + a ^{2} )}

\frac{e ^{- x}}{10}

\frac{( a ^{2} - x ^{2} )}{x}

2 y sin (x) - 4

\frac{1}{( v ^{2} + 3 )}

What is the integral of log_{10}(a^2+x^2) ?
The integral of log_{10}(a^2+x^2) is xlog_{10}(a^2+x^2)+2/(ln(10))aarctan(x/a)-(2x)/(ln(10))+C