Calculus Examples

$\int_{- 1.6}^{0.7} {(1 - x)}^{\frac{1}{3}} d x$

Step 1

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $1 - x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$0 + \frac{d}{d x} [- x]$

Step 1.1.3

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

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$0 - \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 1.2

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

$u_{lower} = 1 - - 1.6$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 1$ by $- 1.6$ .

$u_{lower} = 1 + 1.6$

Step 1.3.2

Add $1$ and $1.6$ .

$u_{lower} = 2.6$

Step 1.4

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

$u_{upper} = 1 - 1 \cdot 0.7$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 1$ by $0.7$ .

$u_{upper} = 1 - 0.7$

Step 1.5.2

Subtract $0.7$ from $1$ .

$u_{upper} = 0.3$

Step 1.6

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

$u_{lower} = 2.6$

$u_{upper} = 0.3$

Step 1.7

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

$\int_{2.6}^{0.3} - u^{\frac{1}{3}} d u$

Step 2

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

$- \int_{2.6}^{0.3} u^{\frac{1}{3}} d u$

Step 3

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

$- (\frac{3}{4} u^{\frac{4}{3}}]_{2.6}^{0.3})$

Step 4

Combine $\frac{3}{4}$ and $u^{\frac{4}{3}}$ .

$- (\frac{3 u^{\frac{4}{3}}}{4}]_{2.6}^{0.3})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{3 u^{\frac{4}{3}}}{4}$ at $0.3$ and at $2.6$ .

$- ((\frac{3 \cdot {0.3}^{\frac{4}{3}}}{4}) - \frac{3 \cdot {2.6}^{\frac{4}{3}}}{4})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $0.3$ to the power of $\frac{4}{3}$ .

$- (\frac{3 \cdot 0.20082988}{4} - \frac{3 \cdot {2.6}^{\frac{4}{3}}}{4})$

Step 5.2.2

Multiply $3$ by $0.20082988$ .

$- (\frac{0.60248965}{4} - \frac{3 \cdot {2.6}^{\frac{4}{3}}}{4})$

Step 5.2.3

Raise $2.6$ to the power of $\frac{4}{3}$ .

$- (\frac{0.60248965}{4} - \frac{3 \cdot 3.57517905}{4})$

Step 5.2.4

Multiply $3$ by $3.57517905$ .

$- (\frac{0.60248965}{4} - \frac{10.72553716}{4})$

Step 5.2.5

Combine the numerators over the common denominator.

$- \frac{0.60248965 - 10.72553716}{4}$

Step 5.2.6

Subtract $10.72553716$ from $0.60248965$ .

$- \frac{- 10.1230475}{4}$

Step 5.2.7

Move the negative in front of the fraction.

$- - \frac{10.1230475}{4}$

Step 5.2.8

Multiply $- 1$ by $- 1$ .

$1 (\frac{10.1230475}{4})$

Step 5.2.9

Multiply $\frac{10.1230475}{4}$ by $1$ .

$\frac{10.1230475}{4}$

Step 6

Divide $10.1230475$ by $4$ .

$2.53076187$

Step 7