Calculus Examples

$\int \frac{8 x^{4} - 21 x^{3} + 10 x^{2}}{x} d x$

Step 1

Split the fraction into multiple fractions.

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Step 1.1

Factor $x^{2}$ out of $8 x^{4} - 21 x^{3} + 10 x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $8 x^{4}$ .

$\int \frac{x^{2} (8 x^{2}) - 21 x^{3} + 10 x^{2}}{x} d x$

Step 1.1.2

Factor $x^{2}$ out of $- 21 x^{3}$ .

$\int \frac{x^{2} (8 x^{2}) + x^{2} (- 21 x) + 10 x^{2}}{x} d x$

Step 1.1.3

Factor $x^{2}$ out of $10 x^{2}$ .

$\int \frac{x^{2} (8 x^{2}) + x^{2} (- 21 x) + x^{2} \cdot 10}{x} d x$

Step 1.1.4

Factor $x^{2}$ out of $x^{2} (8 x^{2}) + x^{2} (- 21 x)$ .

$\int \frac{x^{2} (8 x^{2} - 21 x) + x^{2} \cdot 10}{x} d x$

Step 1.1.5

Factor $x^{2}$ out of $x^{2} (8 x^{2} - 21 x) + x^{2} \cdot 10$ .

$\int \frac{x^{2} (8 x^{2} - 21 x + 10)}{x} d x$

Step 1.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.2.1

Factor $x$ out of $x^{2} (8 x^{2} - 21 x + 10)$ .

$\int \frac{x (x (8 x^{2} - 21 x + 10))}{x} d x$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

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

$\int \frac{x (x (8 x^{2} - 21 x + 10))}{x^{1}} d x$

Step 1.2.2.2

Factor $x$ out of $x^{1}$ .

$\int \frac{x (x (8 x^{2} - 21 x + 10))}{x \cdot 1} d x$

Step 1.2.2.3

Cancel the common factor.

$\int \frac{x (x (8 x^{2} - 21 x + 10))}{x \cdot 1} d x$

Step 1.2.2.4

Rewrite the expression.

$\int \frac{x (8 x^{2} - 21 x + 10)}{1} d x$

Step 1.2.2.5

Divide $x (8 x^{2} - 21 x + 10)$ by $1$ .

$\int x (8 x^{2} - 21 x + 10) d x$

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

$\int x (8 x^{2} - 21 x) + x \cdot 10 d x$

Step 2.2

Apply the distributive property.

$\int x (8 x^{2}) + x (- 21 x) + x \cdot 10 d x$

Step 2.3

Reorder $x$ and $8$ .

$\int 8 \cdot x \cdot x^{2} + x (- 21 x) + x \cdot 10 d x$

Step 2.4

Reorder $x$ and $- 21$ .

$\int 8 \cdot x \cdot x^{2} - 21 \cdot x \cdot x + x \cdot 10 d x$

Step 2.5

Reorder $x$ and $10$ .

$\int 8 \cdot x \cdot x^{2} - 21 \cdot x \cdot x + 10 \cdot x d x$

Step 2.6

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

$\int 8 (x^{1} x^{2}) - 21 x \cdot x + 10 x d x$

Step 2.7

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

$\int 8 x^{1 + 2} - 21 x \cdot x + 10 x d x$

Step 2.8

Add $1$ and $2$ .

$\int 8 x^{3} - 21 x \cdot x + 10 x d x$

Step 2.9

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

$\int 8 x^{3} - 21 (x^{1} x) + 10 x d x$

Step 2.10

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

$\int 8 x^{3} - 21 (x^{1} x^{1}) + 10 x d x$

Step 2.11

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

$\int 8 x^{3} - 21 x^{1 + 1} + 10 x d x$

Step 2.12

Add $1$ and $1$ .

$\int 8 x^{3} - 21 x^{2} + 10 x d x$

Step 3

Split the single integral into multiple integrals.

$\int 8 x^{3} d x + \int - 21 x^{2} d x + \int 10 x d x$

Step 4

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int x^{3} d x + \int - 21 x^{2} d x + \int 10 x d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$8 (\frac{1}{4} x^{4} + C) + \int - 21 x^{2} d x + \int 10 x d x$

Step 6

Since $- 21$ is constant with respect to $x$ , move $- 21$ out of the integral.

$8 (\frac{1}{4} x^{4} + C) - 21 \int x^{2} d x + \int 10 x d x$

Step 7

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

$8 (\frac{1}{4} x^{4} + C) - 21 (\frac{1}{3} x^{3} + C) + \int 10 x d x$

Step 8

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$8 (\frac{1}{4} x^{4} + C) - 21 (\frac{1}{3} x^{3} + C) + 10 \int x d x$

Step 9

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

$8 (\frac{1}{4} x^{4} + C) - 21 (\frac{1}{3} x^{3} + C) + 10 (\frac{1}{2} x^{2} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$2 x^{4} - 7 x^{3} + 10 (\frac{1}{2} x^{2}) + C$

Step 10.2

Simplify.

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Step 10.2.1

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

$2 x^{4} - 7 x^{3} + 10 \frac{x^{2}}{2} + C$

Step 10.2.2

Combine $10$ and $\frac{x^{2}}{2}$ .

$2 x^{4} - 7 x^{3} + \frac{10 x^{2}}{2} + C$

Step 10.2.3

Cancel the common factor of $10$ and $2$ .

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Step 10.2.3.1

Factor $2$ out of $10 x^{2}$ .

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

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $2$ out of $2$ .

$2 x^{4} - 7 x^{3} + \frac{2 (5 x^{2})}{2 (1)} + C$

Step 10.2.3.2.2

Cancel the common factor.

$2 x^{4} - 7 x^{3} + \frac{2 (5 x^{2})}{2 \cdot 1} + C$

Step 10.2.3.2.3

Rewrite the expression.

$2 x^{4} - 7 x^{3} + \frac{5 x^{2}}{1} + C$

Step 10.2.3.2.4

Divide $5 x^{2}$ by $1$ .

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