Calculus Examples

$\int \frac{x^{4} - x^{3}}{x^{2}} d x$

Step 1

Simplify.

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Step 1.1

Factor $x^{3}$ out of $x^{4} - x^{3}$ .

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Step 1.1.1

Factor $x^{3}$ out of $x^{4}$ .

$\int \frac{x^{3} x - x^{3}}{x^{2}} d x$

Step 1.1.2

Factor $x^{3}$ out of $- x^{3}$ .

$\int \frac{x^{3} x + x^{3} \cdot - 1}{x^{2}} d x$

Step 1.1.3

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot - 1$ .

$\int \frac{x^{3} (x - 1)}{x^{2}} d x$

Step 1.2

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 1.2.1

Factor $x^{2}$ out of $x^{3} (x - 1)$ .

$\int \frac{x^{2} (x (x - 1))}{x^{2}} d x$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Multiply by $1$ .

$\int \frac{x^{2} (x (x - 1))}{x^{2} \cdot 1} d x$

Step 1.2.2.2

Cancel the common factor.

$\int \frac{x^{2} (x (x - 1))}{x^{2} \cdot 1} d x$

Step 1.2.2.3

Rewrite the expression.

$\int \frac{x (x - 1)}{1} d x$

Step 1.2.2.4

Divide $x (x - 1)$ by $1$ .

$\int x (x - 1) d x$

Step 2

Multiply $x (x - 1)$ .

$\int x \cdot x + x \cdot - 1 d x$

Step 3

Simplify.

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Step 3.1

Raise $x$ to the power of $1$ .

$\int x^{1} x + x \cdot - 1 d x$

Step 3.2

Raise $x$ to the power of $1$ .

$\int x^{1} x^{1} + x \cdot - 1 d x$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{1 + 1} + x \cdot - 1 d x$

Step 3.4

Add $1$ and $1$ .

$\int x^{2} + x \cdot - 1 d x$

Step 3.5

Move $- 1$ to the left of $x$ .

$\int x^{2} - 1 \cdot x d x$

Step 3.6

Rewrite $- 1 x$ as $- x$ .

$\int x^{2} - x d x$

Step 4

Split the single integral into multiple integrals.

$\int x^{2} d x + \int - x d x$

Step 5

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

$\frac{1}{3} x^{3} + C + \int - x d x$

Step 6

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

$\frac{1}{3} x^{3} + C - \int x d x$

Step 7

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

$\frac{1}{3} x^{3} + C - (\frac{1}{2} x^{2} + C)$

Step 8

Simplify.

$\frac{1}{3} x^{3} - \frac{1}{2} x^{2} + C$