What is the integral from 0 to 1 of xe^{6x} ?

The integral from 0 to 1 of xe^{6x} is (5e^6+1)/(36)

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integral from 0 to 1 of xe^{6x}

\int_{0}^{1} x e^{6 x} d x

\frac{5 e ^{6} + 1}{36}

Decimal Notation

56.05955 \dots

Solution steps

\int_{0}^{1} x e^{6 x} d x

Apply u-substitution

: \int_{0}^{6} \frac{e ^{u} u}{36} d u

= \int_{0}^{6} \frac{e ^{u} u}{36} d u

Take the constant out:

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

= \frac{1}{36} \cdot \int_{0}^{6} e^{u} u d u

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{0}^{6}

= \frac{1}{36} [e^{u} u - \int e^{u} d u]_{0}^{6}

\int e^{u} d u = e^{u}

= \frac{1}{36} [e^{u} u - e^{u}]_{0}^{6}

Compute the boundaries:

5 e^{6} + 1

= \frac{1}{36} (5 e^{6} + 1)

Simplify

= \frac{5 e ^{6} + 1}{36}

\int y - 1 d y

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = 2 e^{x}

\int 5 e^{6 x} cos (e^{6 x}) d x

y^{^{''}} + 2 y^{^{'}} + 5 y = 4 cos (2 t) e^{- t}

\frac{\partial}{\partial x} (sin (x^{2} - 9 x y))

What is the integral from 0 to 1 of xe^{6x} ?
The integral from 0 to 1 of xe^{6x} is (5e^6+1)/(36)