Calculus Examples

$\int_{1}^{2} t^{5} - t^{3} + 1 d t$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{2} t^{5} d t + \int_{1}^{2} - t^{3} d t + \int_{1}^{2} d t$

Step 2

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

$\frac{1}{6} t^{6}]_{1}^{2} + \int_{1}^{2} - t^{3} d t + \int_{1}^{2} d t$

Step 3

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$\frac{1}{6} t^{6}]_{1}^{2} - \int_{1}^{2} t^{3} d t + \int_{1}^{2} d t$

Step 4

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

$\frac{1}{6} t^{6}]_{1}^{2} - (\frac{1}{4} t^{4}]_{1}^{2}) + \int_{1}^{2} d t$

Step 5

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

$\frac{1}{6} t^{6}]_{1}^{2} - (\frac{t^{4}}{4}]_{1}^{2}) + \int_{1}^{2} d t$

Step 6

Apply the constant rule.

$\frac{1}{6} t^{6}]_{1}^{2} - (\frac{t^{4}}{4}]_{1}^{2}) + t]_{1}^{2}$

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{1}{6} t^{6}]_{1}^{2}$ and $t]_{1}^{2}$ .

$\frac{1}{6} t^{6} + t]_{1}^{2} - (\frac{t^{4}}{4}]_{1}^{2})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{1}{6} t^{6} + t$ at $2$ and at $1$ .

$(\frac{1}{6} \cdot 2^{6} + 2) - (\frac{1}{6} \cdot 1^{6} + 1) - (\frac{t^{4}}{4}]_{1}^{2})$

Step 7.2.2

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

$(\frac{1}{6} \cdot 2^{6} + 2) - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raise $2$ to the power of $6$ .

$\frac{1}{6} \cdot 64 + 2 - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.2

Combine $\frac{1}{6}$ and $64$ .

$\frac{64}{6} + 2 - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.3

Cancel the common factor of $64$ and $6$ .

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Step 7.2.3.3.1

Factor $2$ out of $64$ .

$\frac{2 (32)}{6} + 2 - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 32}{2 \cdot 3} + 2 - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.3.2.2

Cancel the common factor.

$\frac{2 \cdot 32}{2 \cdot 3} + 2 - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.3.2.3

Rewrite the expression.

$\frac{32}{3} + 2 - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{32}{3} + 2 \cdot \frac{3}{3} - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.5

Combine $2$ and $\frac{3}{3}$ .

$\frac{32}{3} + \frac{2 \cdot 3}{3} - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.6

Combine the numerators over the common denominator.

$\frac{32 + 2 \cdot 3}{3} - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.7

Simplify the numerator.

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Step 7.2.3.7.1

Multiply $2$ by $3$ .

$\frac{32 + 6}{3} - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.7.2

Add $32$ and $6$ .

$\frac{38}{3} - (\frac{1}{6} \cdot 1^{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.8

One to any power is one.

$\frac{38}{3} - (\frac{1}{6} \cdot 1 + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.9

Multiply $\frac{1}{6}$ by $1$ .

$\frac{38}{3} - (\frac{1}{6} + 1) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.10

Write $1$ as a fraction with a common denominator.

$\frac{38}{3} - (\frac{1}{6} + \frac{6}{6}) - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.11

Combine the numerators over the common denominator.

$\frac{38}{3} - \frac{1 + 6}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.12

Add $1$ and $6$ .

$\frac{38}{3} - \frac{7}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.13

To write $\frac{38}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{38}{3} \cdot \frac{2}{2} - \frac{7}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.14

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.14.1

Multiply $\frac{38}{3}$ by $\frac{2}{2}$ .

$\frac{38 \cdot 2}{3 \cdot 2} - \frac{7}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.14.2

Multiply $3$ by $2$ .

$\frac{38 \cdot 2}{6} - \frac{7}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.15

Combine the numerators over the common denominator.

$\frac{38 \cdot 2 - 7}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.16

Simplify the numerator.

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Step 7.2.3.16.1

Multiply $38$ by $2$ .

$\frac{76 - 7}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.16.2

Subtract $7$ from $76$ .

$\frac{69}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.17

Cancel the common factor of $69$ and $6$ .

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Step 7.2.3.17.1

Factor $3$ out of $69$ .

$\frac{3 (23)}{6} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.17.2

Cancel the common factors.

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Step 7.2.3.17.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 23}{3 \cdot 2} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.17.2.2

Cancel the common factor.

$\frac{3 \cdot 23}{3 \cdot 2} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.17.2.3

Rewrite the expression.

$\frac{23}{2} - ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 7.2.3.18

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

$\frac{23}{2} - (\frac{16}{4} - \frac{1^{4}}{4})$

Step 7.2.3.19

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

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Step 7.2.3.19.1

Factor $4$ out of $16$ .

$\frac{23}{2} - (\frac{4 \cdot 4}{4} - \frac{1^{4}}{4})$

Step 7.2.3.19.2

Cancel the common factors.

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Step 7.2.3.19.2.1

Factor $4$ out of $4$ .

$\frac{23}{2} - (\frac{4 \cdot 4}{4 (1)} - \frac{1^{4}}{4})$

Step 7.2.3.19.2.2

Cancel the common factor.

$\frac{23}{2} - (\frac{4 \cdot 4}{4 \cdot 1} - \frac{1^{4}}{4})$

Step 7.2.3.19.2.3

Rewrite the expression.

$\frac{23}{2} - (\frac{4}{1} - \frac{1^{4}}{4})$

Step 7.2.3.19.2.4

Divide $4$ by $1$ .

$\frac{23}{2} - (4 - \frac{1^{4}}{4})$

Step 7.2.3.20

One to any power is one.

$\frac{23}{2} - (4 - \frac{1}{4})$

Step 7.2.3.21

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

$\frac{23}{2} - (4 \cdot \frac{4}{4} - \frac{1}{4})$

Step 7.2.3.22

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

$\frac{23}{2} - (\frac{4 \cdot 4}{4} - \frac{1}{4})$

Step 7.2.3.23

Combine the numerators over the common denominator.

$\frac{23}{2} - \frac{4 \cdot 4 - 1}{4}$

Step 7.2.3.24

Simplify the numerator.

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Step 7.2.3.24.1

Multiply $4$ by $4$ .

$\frac{23}{2} - \frac{16 - 1}{4}$

Step 7.2.3.24.2

Subtract $1$ from $16$ .

$\frac{23}{2} - \frac{15}{4}$

Step 7.2.3.25

To write $\frac{23}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{23}{2} \cdot \frac{2}{2} - \frac{15}{4}$

Step 7.2.3.26

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.26.1

Multiply $\frac{23}{2}$ by $\frac{2}{2}$ .

$\frac{23 \cdot 2}{2 \cdot 2} - \frac{15}{4}$

Step 7.2.3.26.2

Multiply $2$ by $2$ .

$\frac{23 \cdot 2}{4} - \frac{15}{4}$

Step 7.2.3.27

Combine the numerators over the common denominator.

$\frac{23 \cdot 2 - 15}{4}$

Step 7.2.3.28

Simplify the numerator.

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Step 7.2.3.28.1

Multiply $23$ by $2$ .

$\frac{46 - 15}{4}$

Step 7.2.3.28.2

Subtract $15$ from $46$ .

$\frac{31}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{31}{4}$

Decimal Form:

$7.75$

Mixed Number Form:

$7 \frac{3}{4}$

Step 9