Calculus Examples

$\int_{0}^{x} (1 + 2 t + 3 t^{2} + 4 t^{3}) d t$

Step 1

Remove parentheses.

$\int_{0}^{x} 1 + 2 t + 3 t^{2} + 4 t^{3} d t$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{x} d t + \int_{0}^{x} 2 t d t + \int_{0}^{x} 3 t^{2} d t + \int_{0}^{x} 4 t^{3} d t$

Step 3

Apply the constant rule.

$t]_{0}^{x} + \int_{0}^{x} 2 t d t + \int_{0}^{x} 3 t^{2} d t + \int_{0}^{x} 4 t^{3} d t$

Step 4

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

$t]_{0}^{x} + 2 \int_{0}^{x} t d t + \int_{0}^{x} 3 t^{2} d t + \int_{0}^{x} 4 t^{3} d t$

Step 5

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

$t]_{0}^{x} + 2 (\frac{1}{2} t^{2}]_{0}^{x}) + \int_{0}^{x} 3 t^{2} d t + \int_{0}^{x} 4 t^{3} d t$

Step 6

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

$t]_{0}^{x} + 2 (\frac{t^{2}}{2}]_{0}^{x}) + \int_{0}^{x} 3 t^{2} d t + \int_{0}^{x} 4 t^{3} d t$

Step 7

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

$t]_{0}^{x} + 2 (\frac{t^{2}}{2}]_{0}^{x}) + 3 \int_{0}^{x} t^{2} d t + \int_{0}^{x} 4 t^{3} d t$

Step 8

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

$t]_{0}^{x} + 2 (\frac{t^{2}}{2}]_{0}^{x}) + 3 (\frac{1}{3} t^{3}]_{0}^{x}) + \int_{0}^{x} 4 t^{3} d t$

Step 9

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

$t]_{0}^{x} + 2 (\frac{t^{2}}{2}]_{0}^{x}) + 3 (\frac{t^{3}}{3}]_{0}^{x}) + \int_{0}^{x} 4 t^{3} d t$

Step 10

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

$t]_{0}^{x} + 2 (\frac{t^{2}}{2}]_{0}^{x}) + 3 (\frac{t^{3}}{3}]_{0}^{x}) + 4 \int_{0}^{x} t^{3} d t$

Step 11

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

$t]_{0}^{x} + 2 (\frac{t^{2}}{2}]_{0}^{x}) + 3 (\frac{t^{3}}{3}]_{0}^{x}) + 4 (\frac{1}{4} t^{4}]_{0}^{x})$

Step 12

Simplify the answer.

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Step 12.1

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

$t]_{0}^{x} + 2 (\frac{t^{2}}{2}]_{0}^{x}) + 3 (\frac{t^{3}}{3}]_{0}^{x}) + 4 (\frac{t^{4}}{4}]_{0}^{x})$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate $t$ at $x$ and at $0$ .

$(x) + 0 + 2 (\frac{t^{2}}{2}]_{0}^{x}) + 3 (\frac{t^{3}}{3}]_{0}^{x}) + 4 (\frac{t^{4}}{4}]_{0}^{x})$

Step 12.2.2

Evaluate $\frac{t^{2}}{2}$ at $x$ and at $0$ .

$x + 0 + 2 (\frac{x^{2}}{2} - \frac{0^{2}}{2}) + 3 (\frac{t^{3}}{3}]_{0}^{x}) + 4 (\frac{t^{4}}{4}]_{0}^{x})$

Step 12.2.3

Evaluate $\frac{t^{3}}{3}$ at $x$ and at $0$ .

$x + 0 + 2 (\frac{x^{2}}{2} - \frac{0^{2}}{2}) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{t^{4}}{4}]_{0}^{x})$

Step 12.2.4

Evaluate $\frac{t^{4}}{4}$ at $x$ and at $0$ .

$x + 0 + 2 (\frac{x^{2}}{2} - \frac{0^{2}}{2}) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5

Simplify.

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Step 12.2.5.1

Add $x$ and $0$ .

$x + 2 (\frac{x^{2}}{2} - \frac{0^{2}}{2}) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.2

Raising $0$ to any positive power yields $0$ .

$x + 2 (\frac{x^{2}}{2} - \frac{0}{2}) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.3

Cancel the common factor of $0$ and $2$ .

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Step 12.2.5.3.1

Factor $2$ out of $0$ .

$x + 2 (\frac{x^{2}}{2} - \frac{2 (0)}{2}) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.3.2

Cancel the common factors.

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Step 12.2.5.3.2.1

Factor $2$ out of $2$ .

$x + 2 (\frac{x^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.3.2.2

Cancel the common factor.

$x + 2 (\frac{x^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.3.2.3

Rewrite the expression.

$x + 2 (\frac{x^{2}}{2} - \frac{0}{1}) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.3.2.4

Divide $0$ by $1$ .

$x + 2 (\frac{x^{2}}{2} - 0) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.4

Multiply $- 1$ by $0$ .

$x + 2 (\frac{x^{2}}{2} + 0) + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.5

Add $\frac{x^{2}}{2}$ and $0$ .

$x + 2 \frac{x^{2}}{2} + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.6

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

$x + \frac{2 x^{2}}{2} + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.7

Cancel the common factor of $2$ .

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Step 12.2.5.7.1

Cancel the common factor.

$x + \frac{2 x^{2}}{2} + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.7.2

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

$x + x^{2} + 3 (\frac{x^{3}}{3} - \frac{0^{3}}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.8

Raising $0$ to any positive power yields $0$ .

$x + x^{2} + 3 (\frac{x^{3}}{3} - \frac{0}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.9

Cancel the common factor of $0$ and $3$ .

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Step 12.2.5.9.1

Factor $3$ out of $0$ .

$x + x^{2} + 3 (\frac{x^{3}}{3} - \frac{3 (0)}{3}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.9.2

Cancel the common factors.

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Step 12.2.5.9.2.1

Factor $3$ out of $3$ .

$x + x^{2} + 3 (\frac{x^{3}}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.9.2.2

Cancel the common factor.

$x + x^{2} + 3 (\frac{x^{3}}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.9.2.3

Rewrite the expression.

$x + x^{2} + 3 (\frac{x^{3}}{3} - \frac{0}{1}) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.9.2.4

Divide $0$ by $1$ .

$x + x^{2} + 3 (\frac{x^{3}}{3} - 0) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.10

Multiply $- 1$ by $0$ .

$x + x^{2} + 3 (\frac{x^{3}}{3} + 0) + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.11

Add $\frac{x^{3}}{3}$ and $0$ .

$x + x^{2} + 3 \frac{x^{3}}{3} + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.12

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

$x + x^{2} + \frac{3 x^{3}}{3} + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.13

Cancel the common factor of $3$ .

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Step 12.2.5.13.1

Cancel the common factor.

$x + x^{2} + \frac{3 x^{3}}{3} + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.13.2

Divide $x^{3}$ by $1$ .

$x + x^{2} + x^{3} + 4 (\frac{x^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.14

Raising $0$ to any positive power yields $0$ .

$x + x^{2} + x^{3} + 4 (\frac{x^{4}}{4} - \frac{0}{4})$

Step 12.2.5.15

Cancel the common factor of $0$ and $4$ .

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Step 12.2.5.15.1

Factor $4$ out of $0$ .

$x + x^{2} + x^{3} + 4 (\frac{x^{4}}{4} - \frac{4 (0)}{4})$

Step 12.2.5.15.2

Cancel the common factors.

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Step 12.2.5.15.2.1

Factor $4$ out of $4$ .

$x + x^{2} + x^{3} + 4 (\frac{x^{4}}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 12.2.5.15.2.2

Cancel the common factor.

$x + x^{2} + x^{3} + 4 (\frac{x^{4}}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 12.2.5.15.2.3

Rewrite the expression.

$x + x^{2} + x^{3} + 4 (\frac{x^{4}}{4} - \frac{0}{1})$

Step 12.2.5.15.2.4

Divide $0$ by $1$ .

$x + x^{2} + x^{3} + 4 (\frac{x^{4}}{4} - 0)$

Step 12.2.5.16

Multiply $- 1$ by $0$ .

$x + x^{2} + x^{3} + 4 (\frac{x^{4}}{4} + 0)$

Step 12.2.5.17

Add $\frac{x^{4}}{4}$ and $0$ .

$x + x^{2} + x^{3} + 4 \frac{x^{4}}{4}$

Step 12.2.5.18

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

$x + x^{2} + x^{3} + \frac{4 x^{4}}{4}$

Step 12.2.5.19

Cancel the common factor of $4$ .

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Step 12.2.5.19.1

Cancel the common factor.

$x + x^{2} + x^{3} + \frac{4 x^{4}}{4}$

Step 12.2.5.19.2

Divide $x^{4}$ by $1$ .

$x + x^{2} + x^{3} + x^{4}$