Calculus Examples

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

Step 1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 2.1

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

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

Step 2.2

Multiply $3$ by $- 1$ .

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

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

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

Step 3.5

Subtract $3$ from $3$ .

$\int x^{0} + x^{2} x^{- 3} - x \cdot x^{- 3} + 1 x^{- 3} d x$

Step 3.6

Anything raised to $0$ is $1$ .

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

Step 3.7

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

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

Step 3.8

Subtract $3$ from $2$ .

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

Step 3.9

Factor out negative.

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

Subtract $3$ from $1$ .

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

Step 3.13

Multiply $x^{- 3}$ by $1$ .

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

Step 3.14

Reorder $1$ and $x^{- 1}$ .

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

Step 3.15

Move $1$ .

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

Step 3.16

Move $1$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

$\ln (| x |) + C + \int - x^{- 2} d x + \int x^{- 3} d x + \int d x$

Step 6

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

$\ln (| x |) + C - \int x^{- 2} d x + \int x^{- 3} d x + \int d x$

Step 7

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

$\ln (| x |) + C - (- x^{- 1} + C) + \int x^{- 3} d x + \int d x$

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Simplify.

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