What is the integral from e^x to e^{2x of}xln(y) ?

The integral from e^x to e^{2x of}xln(y) is x(2xe^{2x}-e^{2x}-xe^x+e^x)

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integral from e^x to e^{2x of}xln(y)

\int_{e^{x}}^{e^{2 x}} x ln (y) d y

x (2 x e^{2 x} - e^{2 x} - x e^{x} + e^{x})

Solution steps

\int_{e^{x}}^{e^{2 x}} x ln (y) d y

Take the constant out:

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

= x \cdot \int_{e^{x}}^{e^{2 x}} ln (y) d y

Apply Integration By Parts

: [y ln (y) - \int 1 d y]_{e^{x}}^{e^{2 x}}

= x [y ln (y) - \int 1 d y]_{e^{x}}^{e^{2 x}}

\int 1 d y = y

= x [y ln (y) - y]_{e^{x}}^{e^{2 x}}

Compute the boundaries:

2 x e^{2 x} - e^{2 x} - x e^{x} + e^{x}

= x (2 x e^{2 x} - e^{2 x} - x e^{x} + e^{x})

\int_{1}^{36} \frac{5}{x} d x

\int_{4}^{5} 3 x d x

\int_{2}^{4} 5^{3 x} d x

\int_{1}^{2} \frac{x ^{2} - 1}{x ^{2}} d x

\int_{- 6}^{2} - x^{2} - 3 x d x

What is the integral from e^x to e^{2x of}xln(y) ?
The integral from e^x to e^{2x of}xln(y) is x(2xe^{2x}-e^{2x}-xe^x+e^x)