What is the integral from 1 to 4 of 2pix(e^x-ln(x)) ?

The integral from 1 to 4 of 2pix(e^x-ln(x)) is 2pi(3e^4-16ln(2)+15/4)

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integral from 1 to 4 of 2pix(e^x-ln(x))

\int_{1}^{4} 2 π x (e^{x} - ln (x)) d x

2 π (3 e^{4} - 16 ln (2) + \frac{15}{4})

Decimal Notation

983.03007 \dots

Solution steps

\int_{1}^{4} 2 π x (e^{x} - ln (x)) d x

Take the constant out:

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

= 2 π \cdot \int_{1}^{4} x (e^{x} - ln (x)) d x

Expand

x (e^{x} - ln (x)) : e^{x} x - x ln (x)

= 2 π \cdot \int_{1}^{4} e^{x} x - x ln (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 π (\int_{1}^{4} e^{x} x d x - \int_{1}^{4} x ln (x) d x)

\int_{1}^{4} e^{x} x d x = 3 e^{4}

\int_{1}^{4} x ln (x) d x = 16 ln (2) - \frac{15}{4}

= 2 π (3 e^{4} - (16 ln (2) - \frac{15}{4}))

Simplify

= 2 π (3 e^{4} - 16 ln (2) + \frac{15}{4})

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What is the integral from 1 to 4 of 2pix(e^x-ln(x)) ?
The integral from 1 to 4 of 2pix(e^x-ln(x)) is 2pi(3e^4-16ln(2)+15/4)