Calculus Examples

$\int u^{- 5} (2 - u^{5} + u^{10}) d u$

Step 1

Expand $u^{- 5} (2 - u^{5} + u^{10})$ .

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Step 1.1

Apply the distributive property.

$\int u^{- 5} (2 - u^{5}) + u^{- 5} u^{10} d u$

Step 1.2

Apply the distributive property.

$\int u^{- 5} \cdot 2 + u^{- 5} (- u^{5}) + u^{- 5} u^{10} d u$

Step 1.3

Reorder $u^{- 5}$ and $2$ .

$\int 2 \cdot u^{- 5} + u^{- 5} (- u^{5}) + u^{- 5} u^{10} d u$

Step 1.4

Reorder $u^{- 5}$ and $- 1$ .

$\int 2 \cdot u^{- 5} - 1 \cdot u^{- 5} u^{5} + u^{- 5} u^{10} d u$

Step 1.5

Factor out negative.

$\int 2 u^{- 5} - (u^{- 5} u^{5}) + u^{- 5} u^{10} d u$

Step 1.6

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

$\int 2 u^{- 5} - u^{- 5 + 5} + u^{- 5} u^{10} d u$

Step 1.7

Add $- 5$ and $5$ .

$\int 2 u^{- 5} - u^{0} + u^{- 5} u^{10} d u$

Step 1.8

Anything raised to $0$ is $1$ .

$\int 2 u^{- 5} - 1 \cdot 1 + u^{- 5} u^{10} d u$

Step 1.9

Multiply $- 1$ by $1$ .

$\int 2 u^{- 5} - 1 + u^{- 5} u^{10} d u$

Step 1.10

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

$\int 2 u^{- 5} - 1 + u^{- 5 + 10} d u$

Step 1.11

Add $- 5$ and $10$ .

$\int 2 u^{- 5} - 1 + u^{5} d u$

Step 1.12

Move $- 1$ .

$\int 2 u^{- 5} + u^{5} - 1 d u$

Step 1.13

Reorder $2 u^{- 5}$ and $u^{5}$ .

$\int u^{5} + 2 u^{- 5} - 1 d u$

Step 2

Split the single integral into multiple integrals.

$\int u^{5} d u + \int 2 u^{- 5} d u + \int - 1 d u$

Step 3

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

$\frac{1}{6} u^{6} + C + \int 2 u^{- 5} d u + \int - 1 d u$

Step 4

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$\frac{1}{6} u^{6} + C + 2 \int u^{- 5} d u + \int - 1 d u$

Step 5

By the Power Rule, the integral of $u^{- 5}$ with respect to $u$ is $- \frac{1}{4} u^{- 4}$ .

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

Step 6

Simplify.

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Step 6.1

Combine $u^{- 4}$ and $\frac{1}{4}$ .

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

Step 6.2

Move $u^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7

Apply the constant rule.

$\frac{1}{6} u^{6} + C + 2 (- \frac{1}{4 u^{4}} + C) - u + C$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{u^{6}}{6} - \frac{1}{2 u^{4}} - u + C$

Step 8.2

Reorder terms.

$\frac{1}{6} u^{6} - \frac{1}{2 u^{4}} - u + C$