Calculus Examples

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

Step 1

Integrate both sides with respect to $x$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $x$ .

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

Step 1.2

Split the single integral into multiple integrals.

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

Step 1.3

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

$\frac{d y}{d x} = \frac{1}{3} x^{3} + C_{1} + \int x d x + \int d x$

Step 1.4

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

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

Step 1.5

Apply the constant rule.

$\frac{d y}{d x} = \frac{1}{3} x^{3} + C_{1} + \frac{1}{2} x^{2} + C_{2} + x + C_{3}$

Step 1.6

Simplify.

$\frac{d y}{d x} = \frac{1}{3} x^{3} + \frac{1}{2} x^{2} + x + C_{4}$

Step 2

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d y = \int \frac{1}{3} x^{3} + \frac{1}{2} x^{2} + x + C_{4} d x$

Step 3.2

Apply the constant rule.

$y + C_{5} = \int \frac{1}{3} x^{3} + \frac{1}{2} x^{2} + x + C_{4} d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$y + C_{5} = \int \frac{1}{3} x^{3} d x + \int \frac{1}{2} x^{2} d x + \int x d x + \int C_{4} d x$

Step 3.3.2

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$y + C_{5} = \frac{1}{3} \int x^{3} d x + \int \frac{1}{2} x^{2} d x + \int x d x + \int C_{4} d x$

Step 3.3.3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$y + C_{5} = \frac{1}{3} (\frac{1}{4} x^{4} + C_{6}) + \int \frac{1}{2} x^{2} d x + \int x d x + \int C_{4} d x$

Step 3.3.4

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$y + C_{5} = \frac{1}{3} (\frac{1}{4} x^{4} + C_{6}) + \frac{1}{2} \int x^{2} d x + \int x d x + \int C_{4} d x$

Step 3.3.5

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

$y + C_{5} = \frac{1}{3} (\frac{1}{4} x^{4} + C_{6}) + \frac{1}{2} (\frac{1}{3} x^{3} + C_{7}) + \int x d x + \int C_{4} d x$

Step 3.3.6

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

$y + C_{5} = \frac{1}{3} (\frac{1}{4} x^{4} + C_{6}) + \frac{1}{2} (\frac{1}{3} x^{3} + C_{7}) + \frac{1}{2} x^{2} + C_{8} + \int C_{4} d x$

Step 3.3.7

Apply the constant rule.

$y + C_{5} = \frac{1}{3} (\frac{1}{4} x^{4} + C_{6}) + \frac{1}{2} (\frac{1}{3} x^{3} + C_{7}) + \frac{1}{2} x^{2} + C_{8} + C_{4} x + C_{9}$

Step 3.3.8

Simplify.

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Step 3.3.8.1

Simplify.

$y + C_{5} = \frac{x^{4}}{12} + \frac{x^{3}}{6} + \frac{x^{2}}{2} + C_{4} x + C_{10}$

Step 3.3.8.2

Reorder terms.

$y + C_{5} = \frac{1}{12} x^{4} + \frac{1}{6} x^{3} + \frac{1}{2} x^{2} + C_{4} x + C_{10}$

Step 3.4

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

$y = \frac{1}{12} x^{4} + \frac{1}{6} x^{3} + \frac{1}{2} x^{2} + C x + D$