Calculus Examples

$y e^{y} d y = - \frac{x^{2} + 1}{x} d x$

Step 1

Set up an integral on each side.

$\int y e^{y} d y = \int - \frac{x^{2} + 1}{x} d x$

Step 2

Integrate the left side.

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Step 2.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = e^{y}$ .

$y e^{y} - \int e^{y} d y = \int - \frac{x^{2} + 1}{x} d x$

Step 2.2

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

$y e^{y} - (e^{y} + C_{1}) = \int - \frac{x^{2} + 1}{x} d x$

Step 2.3

Simplify.

$y e^{y} - e^{y} + C_{1} = \int - \frac{x^{2} + 1}{x} d x$

Step 2.4

Reorder terms.

$e^{y} y - e^{y} + C_{1} = \int - \frac{x^{2} + 1}{x} d x$

Step 3

Integrate the right side.

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Step 3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$e^{y} y - e^{y} + C_{1} = - \int \frac{x^{2} + 1}{x} d x$

Step 3.2

Divide $x^{2} + 1$ by $x$ .

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Step 3.2.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$0$

$x^{2}$

$0 x$

$1$

Step 3.2.2

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

					$x$
	$x$	+	$0$		$x^{2}$	+	$0 x$	+	$1$

Step 3.2.3

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	+	$0$

Step 3.2.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0$

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	+	$1$
			-	$x^{2}$	-	$0$

Step 3.2.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	+	$1$
			-	$x^{2}$	-	$0$
						$0$

Step 3.2.6

Pull the next term from the original dividend down into the current dividend.

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	+	$1$
			-	$x^{2}$	-	$0$
						$0$	+	$1$

Step 3.2.7

The final answer is the quotient plus the remainder over the divisor.

$e^{y} y - e^{y} + C_{1} = - \int x + \frac{1}{x} d x$

Step 3.3

Split the single integral into multiple integrals.

$e^{y} y - e^{y} + C_{1} = - (\int x d x + \int \frac{1}{x} d x)$

Step 3.4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$e^{y} y - e^{y} + C_{1} = - (\frac{1}{2} x^{2} + C_{2} + \int \frac{1}{x} d x)$

Step 3.5

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$e^{y} y - e^{y} + C_{1} = - (\frac{1}{2} x^{2} + C_{2} + \ln (| x |) + C_{3})$

Step 3.6

Simplify.

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Step 3.6.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$e^{y} y - e^{y} + C_{1} = - (\frac{x^{2}}{2} + C_{2} + \ln (| x |) + C_{3})$

Step 3.6.2

Simplify.

$e^{y} y - e^{y} + C_{1} = - (\frac{1}{2} x^{2} + \ln (| x |)) + C_{4}$

Step 4

Group the constant of integration on the right side as $C$ .

$e^{y} y - e^{y} = - (\frac{1}{2} x^{2} + \ln (| x |)) + C$