What is the integral of xlog_{10}(x^2+5) ?

The integral of xlog_{10}(x^2+5) is 1/2 x^2log_{10}(x^2+5)-1/(2ln(10))(x^2+5-5ln|x^2+5|)+C

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integral of xlog_{10}(x^2+5)

integral

x lo g_{10} (x^{2} + 5)

\frac{1}{2} x^{2} lo g_{10} (x^{2} + 5) - \frac{1}{2 ln ( 10 )} (x^{2} + 5 - 5 ln x^{2} + 5) + C

Solution steps

\int x lo g_{10} (x^{2} + 5) d x

Apply Integration By Parts

= \frac{1}{2} x^{2} lo g_{10} (x^{2} + 5) - \int \frac{x ^{3}}{ln ( 10 ) ( x ^{2} + 5 )} d x

\int \frac{x ^{3}}{ln ( 10 ) ( x ^{2} + 5 )} d x = \frac{1}{2 ln ( 10 )} (x^{2} + 5 - 5 ln x^{2} + 5)

= \frac{1}{2} x^{2} lo g_{10} (x^{2} + 5) - \frac{1}{2 ln ( 10 )} (x^{2} + 5 - 5 ln x^{2} + 5)

Add a constant to the solution

= \frac{1}{2} x^{2} lo g_{10} (x^{2} + 5) - \frac{1}{2 ln ( 10 )} (x^{2} + 5 - 5 ln x^{2} + 5) + C

What is the integral of xlog_{10}(x^2+5) ?
The integral of xlog_{10}(x^2+5) is 1/2 x^2log_{10}(x^2+5)-1/(2ln(10))(x^2+5-5ln|x^2+5|)+C