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

The integral from 0 to ln(4) of 3(4-xe^x) is 9

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

\int_{0}^{l n (4)} 3 (4 - x e^{x}) d x

9

Solution steps

\int_{0}^{l n (4)} 3 (4 - x e^{x}) d x

Take the constant out:

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

= 3 \cdot \int_{0}^{l n (4)} 4 - 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

= 3 (\int_{0}^{l n (4)} 4 d x - \int_{0}^{l n (4)} x e^{x} d x)

\int_{0}^{l n (4)} 4 d x = 8 ln (2)

\int_{0}^{l n (4)} x e^{x} d x = 8 ln (2) - 3

= 3 (8 ln (2) - (8 ln (2) - 3))

Simplify

= 9

(1 + x^{2}) y^{^{'}} = x y + x^{2} y^{2}

f (x) = \frac{1}{7 x}, at x = 9

y^{^{''}} - 2 x y^{^{'}} = 0

\int_{- 2}^{62} 2 π (y) (2) d y

\int \frac{2}{x ^{2} - 4 x} d x

What is the integral from 0 to ln(4) of 3(4-xe^x) ?
The integral from 0 to ln(4) of 3(4-xe^x) is 9