Calculus Examples

$\frac{7 - 6 x^{3} + 4 x^{6}}{x^{6}}$

Step 1

Move $x^{6}$ out of the denominator by raising it to the $- 1$ power.

$\int (7 - 6 x^{3} + 4 x^{6}) {(x^{6})}^{- 1} d x$

Step 2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 2.1

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

$\int (7 - 6 x^{3} + 4 x^{6}) x^{6 \cdot - 1} d x$

Step 2.2

Multiply $6$ by $- 1$ .

$\int (7 - 6 x^{3} + 4 x^{6}) x^{- 6} d x$

Step 3

Expand $(7 - 6 x^{3} + 4 x^{6}) x^{- 6}$ .

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Step 3.1

Apply the distributive property.

$\int (7 - 6 x^{3}) x^{- 6} + 4 x^{6} x^{- 6} d x$

Step 3.2

Apply the distributive property.

$\int 7 x^{- 6} - 6 x^{3} x^{- 6} + 4 x^{6} x^{- 6} d x$

Step 3.3

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

$\int 7 x^{- 6} - 6 x^{3 - 6} + 4 x^{6} x^{- 6} d x$

Step 3.4

Subtract $6$ from $3$ .

$\int 7 x^{- 6} - 6 x^{- 3} + 4 x^{6} x^{- 6} d x$

Step 3.5

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

$\int 7 x^{- 6} - 6 x^{- 3} + 4 x^{6 - 6} d x$

Step 3.6

Subtract $6$ from $6$ .

$\int 7 x^{- 6} - 6 x^{- 3} + 4 x^{0} d x$

Step 3.7

Anything raised to $0$ is $1$ .

$\int 7 x^{- 6} - 6 x^{- 3} + 4 \cdot 1 d x$

Step 3.8

Multiply $4$ by $1$ .

$\int 7 x^{- 6} - 6 x^{- 3} + 4 d x$

Step 3.9

Reorder $7 x^{- 6}$ and $- 6 x^{- 3}$ .

$\int - 6 x^{- 3} + 7 x^{- 6} + 4 d x$

Step 4

Split the single integral into multiple integrals.

$\int - 6 x^{- 3} d x + \int 7 x^{- 6} d x + \int 4 d x$

Step 5

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$- 6 \int x^{- 3} d x + \int 7 x^{- 6} d x + \int 4 d x$

Step 6

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

$- 6 (- \frac{1}{2} x^{- 2} + C) + \int 7 x^{- 6} d x + \int 4 d x$

Step 7

Simplify.

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Step 7.1

Combine $x^{- 2}$ and $\frac{1}{2}$ .

$- 6 (- \frac{x^{- 2}}{2} + C) + \int 7 x^{- 6} d x + \int 4 d x$

Step 7.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 6 (- \frac{1}{2 x^{2}} + C) + \int 7 x^{- 6} d x + \int 4 d x$

Step 8

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$- 6 (- \frac{1}{2 x^{2}} + C) + 7 \int x^{- 6} d x + \int 4 d x$

Step 9

By the Power Rule, the integral of $x^{- 6}$ with respect to $x$ is $- \frac{1}{5} x^{- 5}$ .

$- 6 (- \frac{1}{2 x^{2}} + C) + 7 (- \frac{1}{5} x^{- 5} + C) + \int 4 d x$

Step 10

Simplify.

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Step 10.1

Combine $x^{- 5}$ and $\frac{1}{5}$ .

$- 6 (- \frac{1}{2 x^{2}} + C) + 7 (- \frac{x^{- 5}}{5} + C) + \int 4 d x$

Step 10.2

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 6 (- \frac{1}{2 x^{2}} + C) + 7 (- \frac{1}{5 x^{5}} + C) + \int 4 d x$

Step 11

Apply the constant rule.

$- 6 (- \frac{1}{2 x^{2}} + C) + 7 (- \frac{1}{5 x^{5}} + C) + 4 x + C$

Step 12

Simplify.

$\frac{3}{x^{2}} - \frac{7}{5 x^{5}} + 4 x + C$