Calculus Examples

$(y^{2} + 1) d x + x^{2} y^{2} d y = 0$

Step 1

Write the problem as a mathematical expression.

$(y^{2} + 1) d x + x^{2} y^{2} d y = 0$

Step 2

Subtract $(y^{2} + 1) d x$ from both sides of the equation.

$x^{2} y^{2} d y = - (y^{2} + 1) d x$

Step 3

Multiply both sides by $\frac{1}{x^{2} (y^{2} + 1)}$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 4.1.1

Factor $x^{2}$ out of $x^{2} y^{2}$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Combine $\frac{1}{y^{2} + 1}$ and $y^{2}$ .

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

Step 4.3

Rewrite using the commutative property of multiplication.

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

Step 4.4

Cancel the common factor of $y^{2} + 1$ .

Tap for more steps...

Step 4.4.1

Move the leading negative in $- \frac{1}{x^{2} (y^{2} + 1)}$ into the numerator.

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

Step 4.4.2

Factor $y^{2} + 1$ out of $x^{2} (y^{2} + 1)$ .

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

Step 4.4.3

Cancel the common factor.

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

Step 4.4.4

Rewrite the expression.

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

Step 4.5

Move the negative in front of the fraction.

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

Step 5

Integrate both sides.

Tap for more steps...

Step 5.1

Set up an integral on each side.

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

Step 5.2

Integrate the left side.

Tap for more steps...

Step 5.2.1

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

Tap for more steps...

Step 5.2.1.1

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

$y^{2}$

$0 y$

$1$

$y^{2}$

$0 y$

$0$

Step 5.2.1.2

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

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

Step 5.2.1.3

Multiply the new quotient term by the divisor.

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

Step 5.2.1.4

The expression needs to be subtracted from the dividend, so change all the signs in $y^{2} + 0 + 1$

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

Step 5.2.1.5

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

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

Step 5.2.1.6

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

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

Step 5.2.2

Split the single integral into multiple integrals.

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

Step 5.2.3

Apply the constant rule.

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

Step 5.2.4

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

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

Step 5.2.5

Simplify the expression.

Tap for more steps...

Step 5.2.5.1

Reorder $y^{2}$ and $1$ .

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

Step 5.2.5.2

Rewrite $1$ as $1^{2}$ .

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

Step 5.2.6

The integral of $\frac{1}{1^{2} + y^{2}}$ with respect to $y$ is $\arctan (y) + C_{2}$ .

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

Step 5.2.7

Simplify.

$y - \arctan (y) + C_{3} = \int - \frac{1}{x^{2}} d x$

Step 5.3

Integrate the right side.

Tap for more steps...

Step 5.3.1

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

$y - \arctan (y) + C_{3} = - \int \frac{1}{x^{2}} d x$

Step 5.3.2

Apply basic rules of exponents.

Tap for more steps...

Step 5.3.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$y - \arctan (y) + C_{3} = - \int {(x^{2})}^{- 1} d x$

Step 5.3.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 5.3.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y - \arctan (y) + C_{3} = - \int x^{2 \cdot - 1} d x$

Step 5.3.2.2.2

Multiply $2$ by $- 1$ .

$y - \arctan (y) + C_{3} = - \int x^{- 2} d x$

Step 5.3.3

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

$y - \arctan (y) + C_{3} = - (- x^{- 1} + C_{4})$

Step 5.3.4

Simplify the answer.

Tap for more steps...

Step 5.3.4.1

Rewrite $- (- x^{- 1} + C_{4})$ as $- - \frac{1}{x} + C_{4}$ .

$y - \arctan (y) + C_{3} = - - \frac{1}{x} + C_{4}$

Step 5.3.4.2

Simplify.

Tap for more steps...

Step 5.3.4.2.1

Multiply $- 1$ by $- 1$ .

$y - \arctan (y) + C_{3} = 1 \frac{1}{x} + C_{4}$

Step 5.3.4.2.2

Multiply $\frac{1}{x}$ by $1$ .

$y - \arctan (y) + C_{3} = \frac{1}{x} + C_{4}$

Step 5.4

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

$y - \arctan (y) = \frac{1}{x} + K$