Calculus Examples

$\int_{1}^{2} (3 x^{2} + 4 x^{3}) d x$

Step 1

Remove parentheses.

$\int_{1}^{2} 3 x^{2} + 4 x^{3} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{2} 3 x^{2} d x + \int_{1}^{2} 4 x^{3} d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{1}^{2} x^{2} d x + \int_{1}^{2} 4 x^{3} d x$

Step 4

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

$3 (\frac{1}{3} x^{3}]_{1}^{2}) + \int_{1}^{2} 4 x^{3} d x$

Step 5

Combine $\frac{1}{3}$ and $x^{3}$ .

$3 (\frac{x^{3}}{3}]_{1}^{2}) + \int_{1}^{2} 4 x^{3} d x$

Step 6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$3 (\frac{x^{3}}{3}]_{1}^{2}) + 4 \int_{1}^{2} x^{3} d x$

Step 7

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

$3 (\frac{x^{3}}{3}]_{1}^{2}) + 4 (\frac{1}{4} x^{4}]_{1}^{2})$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$3 (\frac{x^{3}}{3}]_{1}^{2}) + 4 (\frac{x^{4}}{4}]_{1}^{2})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $1$ .

$3 ((\frac{2^{3}}{3}) - \frac{1^{3}}{3}) + 4 (\frac{x^{4}}{4}]_{1}^{2})$

Step 8.2.2

Evaluate $\frac{x^{4}}{4}$ at $2$ and at $1$ .

$3 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + 4 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 8.2.3

Simplify.

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Step 8.2.3.1

Raise $2$ to the power of $3$ .

$3 (\frac{8}{3} - \frac{1^{3}}{3}) + 4 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 8.2.3.2

One to any power is one.

$3 (\frac{8}{3} - \frac{1}{3}) + 4 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 8.2.3.3

Combine the numerators over the common denominator.

$3 \frac{8 - 1}{3} + 4 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 8.2.3.4

Subtract $1$ from $8$ .

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

Step 8.2.3.5

Combine $3$ and $\frac{7}{3}$ .

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

Step 8.2.3.6

Multiply $3$ by $7$ .

$\frac{21}{3} + 4 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 8.2.3.7

Cancel the common factor of $21$ and $3$ .

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Step 8.2.3.7.1

Factor $3$ out of $21$ .

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

Step 8.2.3.7.2

Cancel the common factors.

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Step 8.2.3.7.2.1

Factor $3$ out of $3$ .

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

Step 8.2.3.7.2.2

Cancel the common factor.

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

Step 8.2.3.7.2.3

Rewrite the expression.

$\frac{7}{1} + 4 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 8.2.3.7.2.4

Divide $7$ by $1$ .

$7 + 4 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 8.2.3.8

Raise $2$ to the power of $4$ .

$7 + 4 (\frac{16}{4} - \frac{1^{4}}{4})$

Step 8.2.3.9

Cancel the common factor of $16$ and $4$ .

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Step 8.2.3.9.1

Factor $4$ out of $16$ .

$7 + 4 (\frac{4 \cdot 4}{4} - \frac{1^{4}}{4})$

Step 8.2.3.9.2

Cancel the common factors.

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Step 8.2.3.9.2.1

Factor $4$ out of $4$ .

$7 + 4 (\frac{4 \cdot 4}{4 (1)} - \frac{1^{4}}{4})$

Step 8.2.3.9.2.2

Cancel the common factor.

$7 + 4 (\frac{4 \cdot 4}{4 \cdot 1} - \frac{1^{4}}{4})$

Step 8.2.3.9.2.3

Rewrite the expression.

$7 + 4 (\frac{4}{1} - \frac{1^{4}}{4})$

Step 8.2.3.9.2.4

Divide $4$ by $1$ .

$7 + 4 (4 - \frac{1^{4}}{4})$

Step 8.2.3.10

One to any power is one.

$7 + 4 (4 - \frac{1}{4})$

Step 8.2.3.11

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$7 + 4 (4 \cdot \frac{4}{4} - \frac{1}{4})$

Step 8.2.3.12

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

$7 + 4 (\frac{4 \cdot 4}{4} - \frac{1}{4})$

Step 8.2.3.13

Combine the numerators over the common denominator.

$7 + 4 \frac{4 \cdot 4 - 1}{4}$

Step 8.2.3.14

Simplify the numerator.

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Step 8.2.3.14.1

Multiply $4$ by $4$ .

$7 + 4 \frac{16 - 1}{4}$

Step 8.2.3.14.2

Subtract $1$ from $16$ .

$7 + 4 (\frac{15}{4})$

Step 8.2.3.15

Combine $4$ and $\frac{15}{4}$ .

$7 + \frac{4 \cdot 15}{4}$

Step 8.2.3.16

Multiply $4$ by $15$ .

$7 + \frac{60}{4}$

Step 8.2.3.17

Cancel the common factor of $60$ and $4$ .

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Step 8.2.3.17.1

Factor $4$ out of $60$ .

$7 + \frac{4 \cdot 15}{4}$

Step 8.2.3.17.2

Cancel the common factors.

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Step 8.2.3.17.2.1

Factor $4$ out of $4$ .

$7 + \frac{4 \cdot 15}{4 (1)}$

Step 8.2.3.17.2.2

Cancel the common factor.

$7 + \frac{4 \cdot 15}{4 \cdot 1}$

Step 8.2.3.17.2.3

Rewrite the expression.

$7 + \frac{15}{1}$

Step 8.2.3.17.2.4

Divide $15$ by $1$ .

$7 + 15$

Step 8.2.3.18

Add $7$ and $15$ .

$22$

Step 9