Calculus Examples

$\int_{15}^{35} (x^{2} x) d x$

Step 1

Remove parentheses.

$\int_{15}^{35} x^{2} x d x$

Step 2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1

Multiply $x^{2}$ by $x$ .

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Step 2.1.1

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

$\int_{15}^{35} x^{2} x^{1} d x$

Step 2.1.2

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

$\int_{15}^{35} x^{2 + 1} d x$

Step 2.2

Add $2$ and $1$ .

$\int_{15}^{35} x^{3} d x$

Step 3

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

$\frac{1}{4} x^{4}]_{15}^{35}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{1}{4} x^{4}$ at $35$ and at $15$ .

$(\frac{1}{4} \cdot 35^{4}) - \frac{1}{4} \cdot 15^{4}$

Step 4.2

Simplify.

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Step 4.2.1

Raise $35$ to the power of $4$ .

$\frac{1}{4} \cdot 1500625 - \frac{1}{4} \cdot 15^{4}$

Step 4.2.2

Combine $\frac{1}{4}$ and $1500625$ .

$\frac{1500625}{4} - \frac{1}{4} \cdot 15^{4}$

Step 4.2.3

Raise $15$ to the power of $4$ .

$\frac{1500625}{4} - \frac{1}{4} \cdot 50625$

Step 4.2.4

Multiply $50625$ by $- 1$ .

$\frac{1500625}{4} - 50625 (\frac{1}{4})$

Step 4.2.5

Combine $- 50625$ and $\frac{1}{4}$ .

$\frac{1500625}{4} + \frac{- 50625}{4}$

Step 4.2.6

Move the negative in front of the fraction.

$\frac{1500625}{4} - \frac{50625}{4}$

Step 4.2.7

Combine the numerators over the common denominator.

$\frac{1500625 - 50625}{4}$

Step 4.2.8

Subtract $50625$ from $1500625$ .

$\frac{1450000}{4}$

Step 4.2.9

Cancel the common factor of $1450000$ and $4$ .

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Step 4.2.9.1

Factor $4$ out of $1450000$ .

$\frac{4 \cdot 362500}{4}$

Step 4.2.9.2

Cancel the common factors.

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Step 4.2.9.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 362500}{4 (1)}$

Step 4.2.9.2.2

Cancel the common factor.

$\frac{4 \cdot 362500}{4 \cdot 1}$

Step 4.2.9.2.3

Rewrite the expression.

$\frac{362500}{1}$

Step 4.2.9.2.4

Divide $362500$ by $1$ .

$362500$

Step 5