Calculus Examples

$\int {(7 x^{3})}^{2} d x$

Step 1

Simplify.

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Step 1.1

Factor $7$ out of $7 x^{3}$ .

$\int {(7 (x^{3}))}^{2} d x$

Step 1.2

Apply the product rule to $7 (x^{3})$ .

$\int 7^{2} {(x^{3})}^{2} d x$

Step 1.3

Raise $7$ to the power of $2$ .

$\int 49 {(x^{3})}^{2} d x$

Step 1.4

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 1.4.1

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

$\int 49 x^{3 \cdot 2} d x$

Step 1.4.2

Multiply $3$ by $2$ .

$\int 49 x^{6} d x$

Step 2

Since $49$ is constant with respect to $x$ , move $49$ out of the integral.

$49 \int x^{6} d x$

Step 3

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$49 (\frac{1}{7} x^{7} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $49 (\frac{1}{7} x^{7} + C)$ as $49 (\frac{1}{7}) x^{7} + C$ .

$49 (\frac{1}{7}) x^{7} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $49$ and $\frac{1}{7}$ .

$\frac{49}{7} x^{7} + C$

Step 4.2.2

Cancel the common factor of $49$ and $7$ .

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Step 4.2.2.1

Factor $7$ out of $49$ .

$\frac{7 \cdot 7}{7} x^{7} + C$

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

Factor $7$ out of $7$ .

$\frac{7 \cdot 7}{7 (1)} x^{7} + C$

Step 4.2.2.2.2

Cancel the common factor.

$\frac{7 \cdot 7}{7 \cdot 1} x^{7} + C$

Step 4.2.2.2.3

Rewrite the expression.

$\frac{7}{1} x^{7} + C$

Step 4.2.2.2.4

Divide $7$ by $1$ .

$7 x^{7} + C$