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

The integral of x-(1/2)^x is (x^2)/2+1/(ln(2)*2^x)+C

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integral of x-(1/2)^x

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

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

Solution steps

\int x - (\frac{1}{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

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

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

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

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

Simplify

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

Add a constant to the solution

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

x \to \infty lim (4^{x} - 1)

x \to 3 lim (\frac{( x ^{2} - 9 )}{2 x - 6})

x \to \infty lim (\frac{2 ^{x}}{3 ^{x - 1}})

\int 2 - x d x

\frac{d}{d x} (cos (\frac{π}{2} x))

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