Calculus Examples

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

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 $x$ .

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

Step 1.2

Split the single integral into multiple integrals.

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

Step 1.3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 1.4

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

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

Step 1.5

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 1.6

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

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

Step 1.7

Apply the constant rule.

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

Step 1.8

Simplify.

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Step 1.8.1

Simplify.

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Step 1.8.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 1.8.1.2

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

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

Step 1.8.2

Simplify.

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

Step 2

Rewrite the equation.

$d y = (x^{4} + x^{3} + x + C_{4}) d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

Apply the constant rule.

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

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

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

Step 3.3.2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Apply the constant rule.

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

Step 3.3.6

Simplify.

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

Step 3.4

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

$y = \frac{1}{5} x^{5} + \frac{1}{4} x^{4} + \frac{1}{2} x^{2} + C x + D$