Iterated Integral

Can anybody do the following integral?

$\int\limits_1^2\int\limits_{x^3}^{x}e^{y/x}dydx$

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

You start by evaluating the inner integral

$\int_1^2 \! \int_{x^3}^{x} \! e^{\frac{y}{x}} \, dy \, dx = \int_1^2 \! [xe^{\frac{y}{x}}]_{x^3}^{x} \, dx = \int_1^2 \! (ex-xe^{x^2}) \, dx$

You then follow by evaluating the outer integral

$\int_1^2 \! (ex-xe^{x^2}) \, dx = [ \frac{e}{2} x^2 - \frac{1}{2} e^{x^2} ]_1^2 = ( 2e - \frac{1}{2} e^{4} ) - ( \frac{1}{2} e - \frac{1}{2} e ) = \boxed{\frac{1}{2} e (4 - e^{3})}$

Cole Coupland - 7 years, 2 months ago

thanks alot ^^

Esraa Ibrahim - 7 years, 2 months ago

Aw... i know the answer but i dont know how to explain it in this website....

Raka Panuntun - 7 years, 2 months ago

thanks...no problem^^

Esraa Ibrahim - 7 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$