Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Factor $x^{2}$ out of $x^{5} + 2 x^{4} - x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $x^{5}$ .

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

Step 1.1.2

Factor $x^{2}$ out of $2 x^{4}$ .

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

Step 1.1.3

Factor $x^{2}$ out of $- x^{2}$ .

$\int \frac{x^{2} x^{3} + x^{2} (2 x^{2}) + x^{2} \cdot - 1}{x^{4}} d x$

Step 1.1.4

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

$\int \frac{x^{2} (x^{3} + 2 x^{2}) + x^{2} \cdot - 1}{x^{4}} d x$

Step 1.1.5

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

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

Expand $(x^{3} + 2 x^{2} - 1) x^{- 2}$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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

Step 3.4

Subtract $2$ from $3$ .

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

Step 3.5

Simplify.

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

Step 3.6

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

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

Step 3.7

Subtract $2$ from $2$ .

$\int x + 2 x^{0} - 1 x^{- 2} d x$

Step 3.8

Anything raised to $0$ is $1$ .

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

Step 3.9

Multiply $2$ by $1$ .

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

Step 3.10

Move $2$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Reorder terms.

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