What is the integral of (x^2+ln(2x)+e^{-2x}) ?

The integral of (x^2+ln(2x)+e^{-2x}) is (x^3)/3+xln(2x)-x-1/2 e^{-2x}+C

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integral of (x^2+ln(2x)+e^{-2x})

\int (x^{2} + ln (2 x) + e^{- 2 x}) d x

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

Solution steps

\int x^{2} + ln (2 x) + e^{- 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^{2} d x + \int ln (2 x) d x + \int e^{- 2 x} d x

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

\int ln (2 x) d x = x ln (2 x) - x

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

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

Add a constant to the solution

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

x \to 0 lim (1 - e^{- x})

\int \frac{( a + x )}{( a - x )} d x

x \to - 10 lim (x^{2} - 11 x + 10)

y = 2 x - 8

x \to 1 lim (\frac{1}{x ( x + 2 )})

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