Calculus Examples

$\int {(7 x)}^{5} v ({(7 x)}^{3} + 1) d x$

Step 1

Simplify.

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Step 1.1

Factor $7$ out of $7 x$ .

$\int {(7 (x))}^{5} v ({(7 x)}^{3} + 1) d x$

Step 1.2

Apply the product rule to $7 (x)$ .

$\int 7^{5} x^{5} v ({(7 x)}^{3} + 1) d x$

Step 1.3

Raise $7$ to the power of $5$ .

$\int 16807 x^{5} v ({(7 x)}^{3} + 1) d x$

Step 1.4

Factor $7$ out of $7 x$ .

$\int 16807 x^{5} v ({(7 (x))}^{3} + 1) d x$

Step 1.5

Apply the product rule to $7 (x)$ .

$\int 16807 x^{5} v (7^{3} x^{3} + 1) d x$

Step 1.6

Raise $7$ to the power of $3$ .

$\int 16807 x^{5} v (343 x^{3} + 1) d x$

Step 2

Since $16807 v$ is constant with respect to $x$ , move $16807 v$ out of the integral.

$16807 v \int x^{5} (343 x^{3} + 1) d x$

Step 3

Multiply $x^{5} (343 x^{3} + 1)$ .

$16807 v \int x^{5} (343 x^{3}) + x^{5} \cdot 1 d x$

Step 4

Simplify.

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Step 4.1

Multiply $x^{5}$ by $x^{3}$ by adding the exponents.

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Step 4.1.1

Move $x^{3}$ .

$16807 v \int x^{3} x^{5} \cdot 343 + x^{5} \cdot 1 d x$

Step 4.1.2

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

$16807 v \int x^{3 + 5} \cdot 343 + x^{5} \cdot 1 d x$

Step 4.1.3

Add $3$ and $5$ .

$16807 v \int x^{8} \cdot 343 + x^{5} \cdot 1 d x$

Step 4.2

Move $343$ to the left of $x^{8}$ .

$16807 v \int 343 \cdot x^{8} + x^{5} \cdot 1 d x$

Step 4.3

Multiply $x^{5}$ by $1$ .

$16807 v \int 343 x^{8} + x^{5} d x$

Step 5

Split the single integral into multiple integrals.

$16807 v (\int 343 x^{8} d x + \int x^{5} d x)$

Step 6

Since $343$ is constant with respect to $x$ , move $343$ out of the integral.

$16807 v (343 \int x^{8} d x + \int x^{5} d x)$

Step 7

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$16807 v (343 (\frac{1}{9} x^{9} + C) + \int x^{5} d x)$

Step 8

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

$16807 v (343 (\frac{1}{9} x^{9} + C) + \frac{1}{6} x^{6} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{9}$ and $x^{9}$ .

$16807 v (343 (\frac{x^{9}}{9} + C) + \frac{1}{6} x^{6} + C)$

Step 9.1.2

Combine $\frac{1}{6}$ and $x^{6}$ .

$16807 v (343 (\frac{x^{9}}{9} + C) + \frac{x^{6}}{6} + C)$

Step 9.2

Simplify.

$16807 v (\frac{343 x^{9}}{9} + \frac{x^{6}}{6}) + C$

Step 9.3

Reorder terms.

$16807 v (\frac{343}{9} x^{9} + \frac{1}{6} x^{6}) + C$