What is the integral from 0 to ln(3) of 2(3-xe^x) ?

The integral from 0 to ln(3) of 2(3-xe^x) is 4

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integral from 0 to ln(3) of 2(3-xe^x)

\int_{0}^{l n (3)} 2 (3 - x e^{x}) d x

4

Solution steps

\int_{0}^{l n (3)} 2 (3 - x e^{x}) d x

Take the constant out:

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

= 2 \cdot \int_{0}^{l n (3)} 3 - x e^{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

= 2 (\int_{0}^{l n (3)} 3 d x - \int_{0}^{l n (3)} x e^{x} d x)

\int_{0}^{l n (3)} 3 d x = 3 ln (3)

\int_{0}^{l n (3)} x e^{x} d x = 3 ln (3) - 2

= 2 (3 ln (3) - (3 ln (3) - 2))

Simplify

= 4

\frac{d}{d x} (x^{4} + x^{3} + x^{2} + x + 1)

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

f (x) = 2 - \frac{4}{x}

\int 4 sin^{2} (2 x) d x

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

What is the integral from 0 to ln(3) of 2(3-xe^x) ?
The integral from 0 to ln(3) of 2(3-xe^x) is 4