Calculus Examples

$\int_{- 1}^{1} 3^{2 x - 1} d x$

Step 1

Let $u = 2 x - 1$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x - 1$ .

$\frac{d}{d x} [2 x - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

Substitute the lower limit in for $x$ in $u = 2 x - 1$ .

$u_{lower} = 2 \cdot - 1 - 1$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $2$ by $- 1$ .

$u_{lower} = - 2 - 1$

Step 1.3.2

Subtract $1$ from $- 2$ .

$u_{lower} = - 3$

Step 1.4

Substitute the upper limit in for $x$ in $u = 2 x - 1$ .

$u_{upper} = 2 \cdot 1 - 1$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $2$ by $1$ .

$u_{upper} = 2 - 1$

Step 1.5.2

Subtract $1$ from $2$ .

$u_{upper} = 1$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - 3$

$u_{upper} = 1$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{- 3}^{1} 3^{u} \cdot \frac{1}{2} d u$

Step 2

Combine $3^{u}$ and $\frac{1}{2}$ .

$\int_{- 3}^{1} \frac{3^{u}}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- 3}^{1} 3^{u} d u$

Step 4

The integral of $3^{u}$ with respect to $u$ is $\frac{3^{u}}{\ln (3)}$ .

$\frac{1}{2} \frac{3^{u}}{\ln (3)}]_{- 3}^{1}$

Step 5

Combine $\frac{1}{2}$ and $\frac{3^{u}}{\ln (3)}]_{- 3}^{1}$ .

$\frac{\frac{3^{u}}{\ln (3)}]_{- 3}^{1}}{2}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{3^{u}}{\ln (3)}$ at $1$ and at $- 3$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Evaluate the exponent.

$\frac{\frac{3}{\ln (3)} - \frac{3^{- 3}}{\ln (3)}}{2}$

Step 6.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 6.2.3

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

$\frac{\frac{3}{\ln (3)} - \frac{\frac{1}{27}}{\ln (3)}}{2}$

Step 6.2.4

Rewrite $\frac{\frac{1}{27}}{\ln (3)}$ as a product.

$\frac{\frac{3}{\ln (3)} - (\frac{1}{27} \cdot \frac{1}{\ln (3)})}{2}$

Step 6.2.5

Multiply $\frac{1}{27}$ by $\frac{1}{\ln (3)}$ .

$\frac{\frac{3}{\ln (3)} - \frac{1}{27 \ln (3)}}{2}$

Step 6.2.6

Multiply $\frac{\frac{3}{\ln (3)} - \frac{1}{27 \ln (3)}}{2}$ by $\frac{\ln (3)}{\ln (3)}$ .

$\frac{\ln (3)}{\ln (3)} \cdot \frac{\frac{3}{\ln (3)} - \frac{1}{27 \ln (3)}}{2}$

Step 6.2.7

Combine.

$\frac{\ln (3) (\frac{3}{\ln (3)} - \frac{1}{27 \ln (3)})}{\ln (3) \cdot 2}$

Step 6.2.8

Apply the distributive property.

$\frac{\ln (3) \frac{3}{\ln (3)} + \ln (3) (- \frac{1}{27 \ln (3)})}{\ln (3) \cdot 2}$

Step 6.2.9

Cancel the common factor of $\ln (3)$ .

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Step 6.2.9.1

Cancel the common factor.

$\frac{\ln (3) \frac{3}{\ln (3)} + \ln (3) (- \frac{1}{27 \ln (3)})}{\ln (3) \cdot 2}$

Step 6.2.9.2

Rewrite the expression.

$\frac{3 + \ln (3) (- \frac{1}{27 \ln (3)})}{\ln (3) \cdot 2}$

Step 6.2.10

Combine $\ln (3)$ and $\frac{1}{27 \ln (3)}$ .

$\frac{3 - \frac{\ln (3)}{27 \ln (3)}}{\ln (3) \cdot 2}$

Step 6.2.11

Cancel the common factor of $\ln (3)$ .

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Step 6.2.11.1

Cancel the common factor.

$\frac{3 - \frac{\ln (3)}{27 \ln (3)}}{\ln (3) \cdot 2}$

Step 6.2.11.2

Rewrite the expression.

$\frac{3 - \frac{1}{27}}{\ln (3) \cdot 2}$

Step 6.2.12

To write $3$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{3 \cdot \frac{27}{27} - \frac{1}{27}}{\ln (3) \cdot 2}$

Step 6.2.13

Combine $3$ and $\frac{27}{27}$ .

$\frac{\frac{3 \cdot 27}{27} - \frac{1}{27}}{\ln (3) \cdot 2}$

Step 6.2.14

Combine the numerators over the common denominator.

$\frac{\frac{3 \cdot 27 - 1}{27}}{\ln (3) \cdot 2}$

Step 6.2.15

Simplify the numerator.

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Step 6.2.15.1

Multiply $3$ by $27$ .

$\frac{\frac{81 - 1}{27}}{\ln (3) \cdot 2}$

Step 6.2.15.2

Subtract $1$ from $81$ .

$\frac{\frac{80}{27}}{\ln (3) \cdot 2}$

Step 6.2.16

Move $2$ to the left of $\ln (3)$ .

$\frac{\frac{80}{27}}{2 \cdot \ln (3)}$

Step 6.2.17

Rewrite $\frac{\frac{80}{27}}{2 \ln (3)}$ as a product.

$\frac{80}{27} \cdot \frac{1}{2 \ln (3)}$

Step 6.2.18

Multiply $\frac{80}{27}$ by $\frac{1}{2 \ln (3)}$ .

$\frac{80}{27 (2 \ln (3))}$

Step 6.2.19

Multiply $2$ by $27$ .

$\frac{80}{54 \ln (3)}$

Step 6.2.20

Cancel the common factor of $80$ and $54$ .

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Step 6.2.20.1

Factor $2$ out of $80$ .

$\frac{2 (40)}{54 \ln (3)}$

Step 6.2.20.2

Cancel the common factors.

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Step 6.2.20.2.1

Factor $2$ out of $54 \ln (3)$ .

$\frac{2 (40)}{2 (27 \ln (3))}$

Step 6.2.20.2.2

Cancel the common factor.

$\frac{2 \cdot 40}{2 (27 \ln (3))}$

Step 6.2.20.2.3

Rewrite the expression.

$\frac{40}{27 \ln (3)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{40}{27 \ln (3)}$

Decimal Form:

$1.34850255 \dots$

Step 8