What is the integral of 1/4 x^2-1/2 ln(x) ?

The integral of 1/4 x^2-1/2 ln(x) is (x^3)/(12)-1/2 (xln(x)-x)+C

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integral of 1/4 x^2-1/2 ln(x)

\int \frac{1}{4} x^{2} - \frac{1}{2} ln (x) d x

\frac{x ^{3}}{12} - \frac{1}{2} (x ln (x) - x) + C

Solution steps

\int \frac{1}{4} x^{2} - \frac{1}{2} ln (x) 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 \frac{1}{4} x^{2} d x - \int \frac{1}{2} ln (x) d x

\int \frac{1}{4} x^{2} d x = \frac{x ^{3}}{12}

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

= \frac{x ^{3}}{12} - \frac{1}{2} (x ln (x) - x)

Add a constant to the solution

= \frac{x ^{3}}{12} - \frac{1}{2} (x ln (x) - x) + C

\int (sin^{3} (e^{x})) (cos (e^{x})) e^{x} d x

\int x^{2} - \frac{3}{x} + cos (x) - \frac{2}{x} d x

\int_{- 2}^{1} x^{- 4} d x

\int \frac{1}{( 1 + 5 x ) 5 x} d x

\frac{d}{d x} (\frac{3}{x ^{2}} - \frac{4}{x ^{3}})

What is the integral of 1/4 x^2-1/2 ln(x) ?
The integral of 1/4 x^2-1/2 ln(x) is (x^3)/(12)-1/2 (xln(x)-x)+C