Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Multiply $2$ by $- 1$ .

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

Step 3

Expand $(2 x^{2} + 4 x - 3) x^{- 2}$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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

Step 3.4

Subtract $2$ from $2$ .

$\int 2 x^{0} + 4 x \cdot x^{- 2} - 3 x^{- 2} d x$

Step 3.5

Anything raised to $0$ is $1$ .

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

Step 3.6

Multiply $2$ by $1$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Subtract $2$ from $1$ .

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

Step 3.10

Reorder $2$ and $4 x^{- 1}$ .

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

Step 3.11

Move $2$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

$4 (\ln (| x |) + C) + \int - 3 x^{- 2} d x + \int 2 d x$

Step 7

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

$4 (\ln (| x |) + C) - 3 \int x^{- 2} d x + \int 2 d x$

Step 8

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

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

Step 9

Apply the constant rule.

$4 (\ln (| x |) + C) - 3 (- x^{- 1} + C) + 2 x + C$

Step 10

Simplify.

$4 \ln (| x |) + \frac{3}{x} + 2 x + C$