Calculus Examples

$\frac{d y}{d x} = 5 x^{2} + \frac{4}{x}$ , $y (0) = 1$

Step 1

Rewrite the equation.

$d y = (5 x^{2} + \frac{4}{x}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

$y + C_{1} = 5 (\frac{1}{3} x^{3} + C_{2}) + 4 (\ln (| x |) + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

$y + C_{1} = 5 (\frac{1}{3}) x^{3} + 4 \ln (| x |) + C_{4}$

Step 2.3.6.2

Combine $5$ and $\frac{1}{3}$ .

$y + C_{1} = \frac{5}{3} x^{3} + 4 \ln (| x |) + C_{4}$

Step 2.4

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

$y = \frac{5}{3} x^{3} + 4 \ln (| x |) + C$

Step 3

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $1$ for $y$ in $y = \frac{5}{3} x^{3} + 4 \ln (| x |) + C$ .

$1 = \frac{5}{3} \cdot 0^{3} + 4 \ln (| 0 |) + C$

Step 4

Solve for $C$ .

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Step 4.1

Rewrite the equation as $\frac{5}{3} \cdot 0^{3} + 4 \ln (| 0 |) + C = 1$ .

$\frac{5}{3} \cdot 0^{3} + 4 \ln (| 0 |) + C = 1$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{5}{3} \cdot 0 + 4 \ln (| 0 |) + C = 1$

Step 4.2.1.2

Multiply $\frac{5}{3}$ by $0$ .

$0 + 4 \ln (| 0 |) + C = 1$

Step 4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$0 + 4 \ln (0) + C = 1$

Step 4.2.2

The equation cannot be solved because it is undefined.

$Undefined$