Calculus Examples

$\int {(a^{\frac{2}{3}} - x^{\frac{2}{3}})}^{3} d x$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Use the Binomial Theorem.

$\int {(a^{\frac{2}{3}})}^{3} + 3 {(a^{\frac{2}{3}})}^{2} (- x^{\frac{2}{3}}) + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2

Simplify each term.

Tap for more steps...

Step 1.2.1

Multiply the exponents in ${(a^{\frac{2}{3}})}^{3}$ .

Tap for more steps...

Step 1.2.1.1

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

$\int a^{\frac{2}{3} \cdot 3} + 3 {(a^{\frac{2}{3}})}^{2} (- x^{\frac{2}{3}}) + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.1.2.1

Cancel the common factor.

$\int a^{\frac{2}{3} \cdot 3} + 3 {(a^{\frac{2}{3}})}^{2} (- x^{\frac{2}{3}}) + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.1.2.2

Rewrite the expression.

$\int a^{2} + 3 {(a^{\frac{2}{3}})}^{2} (- x^{\frac{2}{3}}) + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.2

Rewrite using the commutative property of multiplication.

$\int a^{2} + 3 \cdot - 1 {(a^{\frac{2}{3}})}^{2} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.3

Multiply $3$ by $- 1$ .

$\int a^{2} - 3 {(a^{\frac{2}{3}})}^{2} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.4

Multiply the exponents in ${(a^{\frac{2}{3}})}^{2}$ .

Tap for more steps...

Step 1.2.4.1

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

$\int a^{2} - 3 a^{\frac{2}{3} \cdot 2} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.4.2

Multiply $\frac{2}{3} \cdot 2$ .

Tap for more steps...

Step 1.2.4.2.1

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

$\int a^{2} - 3 a^{\frac{2 \cdot 2}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.4.2.2

Multiply $2$ by $2$ .

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} {(- x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.5

Apply the product rule to $- x^{\frac{2}{3}}$ .

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} ({(- 1)}^{2} {(x^{\frac{2}{3}})}^{2}) + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.6

Rewrite using the commutative property of multiplication.

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

Step 1.2.7

Raise $- 1$ to the power of $2$ .

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

Step 1.2.8

Multiply $3$ by $1$ .

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} {(x^{\frac{2}{3}})}^{2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.9

Multiply the exponents in ${(x^{\frac{2}{3}})}^{2}$ .

Tap for more steps...

Step 1.2.9.1

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

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{2}{3} \cdot 2} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.9.2

Multiply $\frac{2}{3} \cdot 2$ .

Tap for more steps...

Step 1.2.9.2.1

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

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{2 \cdot 2}{3}} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.9.2.2

Multiply $2$ by $2$ .

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{4}{3}} + {(- x^{\frac{2}{3}})}^{3} d x$

Step 1.2.10

Apply the product rule to $- x^{\frac{2}{3}}$ .

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{4}{3}} + {(- 1)}^{3} {(x^{\frac{2}{3}})}^{3} d x$

Step 1.2.11

Raise $- 1$ to the power of $3$ .

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{4}{3}} - {(x^{\frac{2}{3}})}^{3} d x$

Step 1.2.12

Multiply the exponents in ${(x^{\frac{2}{3}})}^{3}$ .

Tap for more steps...

Step 1.2.12.1

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

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{4}{3}} - x^{\frac{2}{3} \cdot 3} d x$

Step 1.2.12.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.12.2.1

Cancel the common factor.

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{4}{3}} - x^{\frac{2}{3} \cdot 3} d x$

Step 1.2.12.2.2

Rewrite the expression.

$\int a^{2} - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} + 3 a^{\frac{2}{3}} x^{\frac{4}{3}} - x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int a^{2} d x + \int - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} d x + \int 3 a^{\frac{2}{3}} x^{\frac{4}{3}} d x + \int - x^{2} d x$

Step 3

Apply the constant rule.

$a^{2} x + C + \int - 3 a^{\frac{4}{3}} x^{\frac{2}{3}} d x + \int 3 a^{\frac{2}{3}} x^{\frac{4}{3}} d x + \int - x^{2} d x$

Step 4

Since $- 3 a^{\frac{4}{3}}$ is constant with respect to $x$ , move $- 3 a^{\frac{4}{3}}$ out of the integral.

$a^{2} x + C - 3 a^{\frac{4}{3}} \int x^{\frac{2}{3}} d x + \int 3 a^{\frac{2}{3}} x^{\frac{4}{3}} d x + \int - x^{2} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$a^{2} x + C - 3 a^{\frac{4}{3}} (\frac{3}{5} x^{\frac{5}{3}} + C) + \int 3 a^{\frac{2}{3}} x^{\frac{4}{3}} d x + \int - x^{2} d x$

Step 6

Since $3 a^{\frac{2}{3}}$ is constant with respect to $x$ , move $3 a^{\frac{2}{3}}$ out of the integral.

$a^{2} x + C - 3 a^{\frac{4}{3}} (\frac{3}{5} x^{\frac{5}{3}} + C) + 3 a^{\frac{2}{3}} \int x^{\frac{4}{3}} d x + \int - x^{2} d x$

Step 7

By the Power Rule, the integral of $x^{\frac{4}{3}}$ with respect to $x$ is $\frac{3}{7} x^{\frac{7}{3}}$ .

$a^{2} x + C - 3 a^{\frac{4}{3}} (\frac{3}{5} x^{\frac{5}{3}} + C) + 3 a^{\frac{2}{3}} (\frac{3}{7} x^{\frac{7}{3}} + C) + \int - x^{2} d x$

Step 8

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

$a^{2} x + C - 3 a^{\frac{4}{3}} (\frac{3}{5} x^{\frac{5}{3}} + C) + 3 a^{\frac{2}{3}} (\frac{3}{7} x^{\frac{7}{3}} + C) - \int x^{2} d x$

Step 9

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

$a^{2} x + C - 3 a^{\frac{4}{3}} (\frac{3}{5} x^{\frac{5}{3}} + C) + 3 a^{\frac{2}{3}} (\frac{3}{7} x^{\frac{7}{3}} + C) - (\frac{1}{3} x^{3} + C)$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify.

$a^{2} x - \frac{9 x^{\frac{5}{3}} a^{\frac{4}{3}}}{5} + \frac{9 a^{\frac{2}{3}} x^{\frac{7}{3}}}{7} - (\frac{1}{3} x^{3}) + C$

Step 10.2

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

$a^{2} x - \frac{9 x^{\frac{5}{3}} a^{\frac{4}{3}}}{5} + \frac{9 a^{\frac{2}{3}} x^{\frac{7}{3}}}{7} - \frac{x^{3}}{3} + C$

Step 11

Reorder terms.

$a^{2} x - \frac{9}{5} x^{\frac{5}{3}} a^{\frac{4}{3}} + \frac{9}{7} a^{\frac{2}{3}} x^{\frac{7}{3}} - \frac{1}{3} x^{3} + C$