What is the integral of 1/2 ((x^2)/2+1/4 cos(2x)) ?

The integral of 1/2 ((x^2)/2+1/4 cos(2x)) is 1/2 ((x^3)/6+1/8 sin(2x))+C

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integral of 1/2 ((x^2)/2+1/4 cos(2x))

\int \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) d x

\frac{1}{2} (\frac{x ^{3}}{6} + \frac{1}{8} sin (2 x)) + C

Solution steps

\int \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{x ^{2}}{2} + \frac{1}{4} cos (2 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

= \frac{1}{2} (\int \frac{x ^{2}}{2} d x + \int \frac{1}{4} cos (2 x) d x)

\int \frac{x ^{2}}{2} d x = \frac{x ^{3}}{6}

\int \frac{1}{4} cos (2 x) d x = \frac{1}{8} sin (2 x)

= \frac{1}{2} (\frac{x ^{3}}{6} + \frac{1}{8} sin (2 x))

Add a constant to the solution

= \frac{1}{2} (\frac{x ^{3}}{6} + \frac{1}{8} sin (2 x)) + C

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What is the integral of 1/2 ((x^2)/2+1/4 cos(2x)) ?
The integral of 1/2 ((x^2)/2+1/4 cos(2x)) is 1/2 ((x^3)/6+1/8 sin(2x))+C