Calculus Examples

$\int_{1}^{8} x^{- \frac{2}{3}} d x$

Step 1

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

$3 x^{\frac{1}{3}}]_{1}^{8}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $3 x^{\frac{1}{3}}$ at $8$ and at $1$ .

$(3 \cdot 8^{\frac{1}{3}}) - 3 \cdot 1^{\frac{1}{3}}$

Step 2.2

Simplify.

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Step 2.2.1

Rewrite $8$ as $2^{3}$ .

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

Step 2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.3

Cancel the common factor of $3$ .

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Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Rewrite the expression.

$3 \cdot 2^{1} - 3 \cdot 1^{\frac{1}{3}}$

Step 2.2.4

Evaluate the exponent.

$3 \cdot 2 - 3 \cdot 1^{\frac{1}{3}}$

Step 2.2.5

Multiply $3$ by $2$ .

$6 - 3 \cdot 1^{\frac{1}{3}}$

Step 2.2.6

One to any power is one.

$6 - 3 \cdot 1$

Step 2.2.7

Multiply $- 3$ by $1$ .

$6 - 3$

Step 2.2.8

Subtract $3$ from $6$ .

$3$

Step 3