Calculus Examples

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

Step 1

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

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} + y d y = \int - (t^{2} + 1) d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y^{2} d y + \int y d y = \int - (t^{2} + 1) d t$

Step 2.2.2

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Multiply $- (t^{2} + 1)$ .

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int - t^{2} - 1 \cdot 1 d t$

Step 2.3.2

Multiply $- 1$ by $1$ .

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int - t^{2} - 1 d t$

Step 2.3.3

Split the single integral into multiple integrals.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = \int - t^{2} d t + \int - 1 d t$

Step 2.3.4

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

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = - \int t^{2} d t + \int - 1 d t$

Step 2.3.5

By the Power Rule, the integral of $t^{2}$ with respect to $t$ is $\frac{1}{3} t^{3}$ .

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = - (\frac{1}{3} t^{3} + C_{4}) + \int - 1 d t$

Step 2.3.6

Apply the constant rule.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = - (\frac{1}{3} t^{3} + C_{4}) - t + C_{5}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Combine $\frac{1}{3}$ and $t^{3}$ .

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = - (\frac{t^{3}}{3} + C_{4}) - t + C_{5}$

Step 2.3.7.2

Simplify.

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} + C_{3} = - \frac{1}{3} t^{3} - t + C_{6}$

Step 2.4

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

$\frac{1}{3} y^{3} + \frac{1}{2} y^{2} = - \frac{1}{3} t^{3} - t + K$