Solve \int _(0)^(1)\int _(3y)^(3)y⋅y⋅e^(x⋅y)dx dy

Evaluate

\int_{0}^{1} \int_{3 y}^{3} y \times y e^{x y} d x d y

Multiply the terms

\int_{0}^{1} \int_{3 y}^{3} y^{2} e^{x y} d x d y

To evaluate the iterated inter,first evaluate the inner integral

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\int_{0}^{1} (y e^{3 y} - y e^{3 y^{2}}) d y

Evaluate the integral

\int (y e^{3 y} - y e^{3 y^{2}}) d y

Use the property of integral \int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

\int y e^{3 y} d y + \int - y e^{3 y^{2}} d y

Evaluate the integral

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\frac{y e ^{3 y}}{3} - \frac{e ^{3 y}}{9} + \int - y e^{3 y^{2}} d y

Evaluate the integral

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\frac{y e ^{3 y}}{3} - \frac{e ^{3 y}}{9} - \frac{e ^{3 y^{2}}}{6}

Simplify the expression

\frac{e ^{3 y} y}{3} - \frac{e ^{3 y}}{9} - \frac{e ^{3 y^{2}}}{6}

Return the limits

(\frac{e ^{3 y} y}{3} - \frac{e ^{3 y}}{9} - \frac{e ^{3 y^{2}}}{6})_{0}^{1}

Solution

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Substitute the values into formula

\frac{e ^{3 \times 1} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1^{2}}}{6} - (\frac{e ^{3 \times 0} \times 0}{3} - \frac{e ^{3 \times 0}}{9} - \frac{e ^{3 \times 0^{2}}}{6})

Any expression multiplied by 0 equals 0

\frac{e ^{3 \times 1} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1^{2}}}{6} - (\frac{e ^{0} \times 0}{3} - \frac{e ^{3 \times 0}}{9} - \frac{e ^{3 \times 0^{2}}}{6})

Evaluate the power

\frac{e ^{3 \times 1} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1^{2}}}{6} - (\frac{1 \times 0}{3} - \frac{e ^{3 \times 0}}{9} - \frac{e ^{3 \times 0^{2}}}{6})

Any expression multiplied by 0 equals 0

\frac{e ^{3 \times 1} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1^{2}}}{6} - (\frac{1 \times 0}{3} - \frac{e ^{0}}{9} - \frac{e ^{3 \times 0^{2}}}{6})

Any expression multiplied by 0 equals 0

\frac{e ^{3 \times 1} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1^{2}}}{6} - (\frac{0}{3} - \frac{e ^{0}}{9} - \frac{e ^{3 \times 0^{2}}}{6})

Evaluate the power

\frac{e ^{3 \times 1} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1^{2}}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{e ^{3 \times 0^{2}}}{6})

Calculate

\frac{e ^{3 \times 1} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1^{2}}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{e ^{3 \times 0}}{6})

Any expression multiplied by 1 remains the same

\frac{e ^{3} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1^{2}}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{e ^{3 \times 0}}{6})

1 raised to any power equals to 1

\frac{e ^{3} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{e ^{3 \times 0}}{6})

Any expression multiplied by 0 equals 0

\frac{e ^{3} \times 1}{3} - \frac{e ^{3 \times 1}}{9} - \frac{e ^{3 \times 1}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{e ^{0}}{6})

Any expression multiplied by 1 remains the same

\frac{e ^{3} \times 1}{3} - \frac{e ^{3}}{9} - \frac{e ^{3 \times 1}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{e ^{0}}{6})

Any expression multiplied by 1 remains the same

\frac{e ^{3} \times 1}{3} - \frac{e ^{3}}{9} - \frac{e ^{3}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{e ^{0}}{6})

Evaluate the power

\frac{e ^{3} \times 1}{3} - \frac{e ^{3}}{9} - \frac{e ^{3}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{1}{6})

Any expression multiplied by 1 remains the same

\frac{e ^{3}}{3} - \frac{e ^{3}}{9} - \frac{e ^{3}}{6} - (\frac{0}{3} - \frac{1}{9} - \frac{1}{6})

Divide the terms

\frac{e ^{3}}{3} - \frac{e ^{3}}{9} - \frac{e ^{3}}{6} - (0 - \frac{1}{9} - \frac{1}{6})

Subtract the numbers

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\frac{e ^{3}}{3} - \frac{e ^{3}}{9} - \frac{e ^{3}}{6} - (- \frac{5}{18})

Subtract the numbers

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\frac{2 e ^{3}}{9} - \frac{e ^{3}}{6} - (- \frac{5}{18})

Subtract the numbers

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\frac{e ^{3}}{18} - (- \frac{5}{18})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

\frac{e ^{3}}{18} + \frac{5}{18}

Add the terms

\frac{e ^{3} + 5}{18}

\frac{e ^{3} + 5}{18}

Alternative Form

\approx 1.393641