Calculus Examples

$\int_{1}^{3 x + 2} \frac{t}{1 + t^{3}} d t$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{3}$ .

$\frac{d}{d x} [\int_{1}^{3 x + 2} \frac{t}{1^{3} + t^{3}} d t]$

Step 1.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = t$ .

$\frac{d}{d x} [\int_{1}^{3 x + 2} \frac{t}{(1 + t) (1^{2} - 1 t + t^{2})} d t]$

Step 1.3

Simplify.

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Step 1.3.1

One to any power is one.

$\frac{d}{d x} [\int_{1}^{3 x + 2} \frac{t}{(1 + t) (1 - 1 t + t^{2})} d t]$

Step 1.3.2

Rewrite $- 1 t$ as $- t$ .

$\frac{d}{d x} [\int_{1}^{3 x + 2} \frac{t}{(1 + t) (1 - t + t^{2})} d t]$

Step 2

Take the derivative of $\int_{1}^{3 x + 2} \frac{t}{(1 + t) (1 - t + t^{2})} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [3 x + 2] \frac{3 x + 2}{(1 + 3 x + 2) (1 - (3 x + 2) + {(3 x + 2)}^{2})}$

Step 3

By the Sum Rule, the derivative of $3 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

$(\frac{d}{d x} [3 x] + \frac{d}{d x} [2]) \frac{3 x + 2}{(1 + 3 x + 2) (1 - (3 x + 2) + {(3 x + 2)}^{2})}$

Step 4

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$(3 \frac{d}{d x} [x] + \frac{d}{d x} [2]) \frac{3 x + 2}{(1 + 3 x + 2) (1 - (3 x + 2) + {(3 x + 2)}^{2})}$

Step 4.2

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

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

Step 4.3

Multiply $3$ by $1$ .

$(3 + \frac{d}{d x} [2]) \frac{3 x + 2}{(1 + 3 x + 2) (1 - (3 x + 2) + {(3 x + 2)}^{2})}$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$(3 + 0) \frac{3 x + 2}{(1 + 3 x + 2) (1 - (3 x + 2) + {(3 x + 2)}^{2})}$

Step 5.2

Combine fractions.

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Step 5.2.1

Add $3$ and $0$ .

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

Step 5.2.2

Add $1$ and $2$ .

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

Step 5.2.3

Combine $3$ and $\frac{3 x + 2}{(3 x + 3) (1 - (3 x + 2) + {(3 x + 2)}^{2})}$ .

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

Step 6

Cancel the common factors.

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Step 6.1

Factor $3$ out of $(3 x + 3) (1 - (3 x + 2) + {(3 x + 2)}^{2})$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Combine terms.

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Step 7.2.1

Multiply $3$ by $- 1$ .

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

Step 7.2.2

Multiply $- 1$ by $2$ .

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

Step 7.2.3

Subtract $2$ from $1$ .

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

Step 7.3

Reorder terms.

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

Step 7.4

Simplify the denominator.

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Step 7.4.1

Rewrite ${(3 x + 2)}^{2}$ as $(3 x + 2) (3 x + 2)$ .

$\frac{3 x + 2}{(- 3 x - 1 + (3 x + 2) (3 x + 2)) (x + 1)}$

Step 7.4.2

Expand $(3 x + 2) (3 x + 2)$ using the FOIL Method.

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Step 7.4.2.1

Apply the distributive property.

$\frac{3 x + 2}{(- 3 x - 1 + 3 x (3 x + 2) + 2 (3 x + 2)) (x + 1)}$

Step 7.4.2.2

Apply the distributive property.

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

Step 7.4.2.3

Apply the distributive property.

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

Step 7.4.3

Simplify and combine like terms.

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Step 7.4.3.1

Simplify each term.

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Step 7.4.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 7.4.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 7.4.3.1.2.1

Move $x$ .

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

Step 7.4.3.1.2.2

Multiply $x$ by $x$ .

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

Step 7.4.3.1.3

Multiply $3$ by $3$ .

$\frac{3 x + 2}{(- 3 x - 1 + 9 x^{2} + 3 x \cdot 2 + 2 (3 x) + 2 \cdot 2) (x + 1)}$

Step 7.4.3.1.4

Multiply $2$ by $3$ .

$\frac{3 x + 2}{(- 3 x - 1 + 9 x^{2} + 6 x + 2 (3 x) + 2 \cdot 2) (x + 1)}$

Step 7.4.3.1.5

Multiply $3$ by $2$ .

$\frac{3 x + 2}{(- 3 x - 1 + 9 x^{2} + 6 x + 6 x + 2 \cdot 2) (x + 1)}$

Step 7.4.3.1.6

Multiply $2$ by $2$ .

$\frac{3 x + 2}{(- 3 x - 1 + 9 x^{2} + 6 x + 6 x + 4) (x + 1)}$

Step 7.4.3.2

Add $6 x$ and $6 x$ .

$\frac{3 x + 2}{(- 3 x - 1 + 9 x^{2} + 12 x + 4) (x + 1)}$

Step 7.4.4

Add $- 3 x$ and $12 x$ .

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

Step 7.4.5

Add $- 1$ and $4$ .

$\frac{3 x + 2}{(9 x + 9 x^{2} + 3) (x + 1)}$

Step 7.4.6

Reorder terms.

$\frac{3 x + 2}{(9 x^{2} + 9 x + 3) (x + 1)}$

Step 7.4.7

Factor $3$ out of $9 x^{2} + 9 x + 3$ .

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Step 7.4.7.1

Factor $3$ out of $9 x^{2}$ .

$\frac{3 x + 2}{(3 (3 x^{2}) + 9 x + 3) (x + 1)}$

Step 7.4.7.2

Factor $3$ out of $9 x$ .

$\frac{3 x + 2}{(3 (3 x^{2}) + 3 (3 x) + 3) (x + 1)}$

Step 7.4.7.3

Factor $3$ out of $3$ .

$\frac{3 x + 2}{(3 (3 x^{2}) + 3 (3 x) + 3 (1)) (x + 1)}$

Step 7.4.7.4

Factor $3$ out of $3 (3 x^{2}) + 3 (3 x)$ .

$\frac{3 x + 2}{(3 (3 x^{2} + 3 x) + 3 (1)) (x + 1)}$

Step 7.4.7.5

Factor $3$ out of $3 (3 x^{2} + 3 x) + 3 (1)$ .

$\frac{3 x + 2}{3 (3 x^{2} + 3 x + 1) (x + 1)}$