What is the integral from 0 to 1 of-1x5^x ?

The integral from 0 to 1 of-1x5^x is -(5ln(5)-4)/(ln^2(5))

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integral from 0 to 1 of-1x5^x

\int_{0}^{1} - 1 x 5^{x} d x

- \frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

Decimal

- 1.56244 \dots

Solution steps

\int_{0}^{1} - 1 \cdot x \cdot 5^{x} d x

Take the constant out:

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

= - \int_{0}^{1} x \cdot 5^{x} d x

Apply Integration By Parts

= - [\frac{5 ^{x} x}{ln ( 5 )} - \int \frac{5 ^{x}}{ln ( 5 )} d x]_{0}^{1}

\int \frac{5 ^{x}}{ln ( 5 )} d x = \frac{5 ^{x}}{ln ^{2} ( 5 )}

= - [\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )}]_{0}^{1}

Compute the boundaries:

\frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

= - \frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

What is the integral from 0 to 1 of-1x5^x ?
The integral from 0 to 1 of-1x5^x is -(5ln(5)-4)/(ln^2(5))