Calculus Examples

$\int_{x}^{2 x} 12 t^{2} - 6 t d t$

Step 1

Split the integral into two integrals where $c$ is some value between $x$ and $2 x$ .

$\frac{d}{d x} [\int_{x}^{c} 12 t^{2} - 6 t d t + \int_{c}^{2 x} 12 t^{2} - 6 t d t]$

Step 2

By the Sum Rule, the derivative of $\int_{x}^{c} 12 t^{2} - 6 t d t + \int_{c}^{2 x} 12 t^{2} - 6 t d t$ with respect to $x$ is $\frac{d}{d x} [\int_{x}^{c} 12 t^{2} - 6 t d t] + \frac{d}{d x} [\int_{c}^{2 x} 12 t^{2} - 6 t d t]$ .

$\frac{d}{d x} [\int_{x}^{c} 12 t^{2} - 6 t d t] + \frac{d}{d x} [\int_{c}^{2 x} 12 t^{2} - 6 t d t]$

Step 3

Swap the bounds of integration.

$\frac{d}{d x} [- \int_{c}^{x} 12 t^{2} - 6 t d t] + \frac{d}{d x} [\int_{c}^{2 x} 12 t^{2} - 6 t d t]$

Step 4

Take the derivative of $- \int_{c}^{x} 12 t^{2} - 6 t d t$ with respect to $x$ using Fundamental Theorem of Calculus.

$- (12 x^{2} - 6 x) + \frac{d}{d x} [\int_{c}^{2 x} 12 t^{2} - 6 t d t]$

Step 5

Take the derivative of $\int_{c}^{2 x} 12 t^{2} - 6 t d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$- (12 x^{2} - 6 x) + \frac{d}{d x} [2 x] (12 {(2 x)}^{2} - 6 (2 x))$

Step 6

Differentiate.

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Step 6.1

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

$- (12 x^{2} - 6 x) + 2 \frac{d}{d x} [x] (12 {(2 x)}^{2} - 6 (2 x))$

Step 6.2

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

$- (12 x^{2} - 6 x) + 2 \cdot 1 (12 {(2 x)}^{2} - 6 (2 x))$

Step 6.3

Simplify with factoring out.

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Step 6.3.1

Multiply $2$ by $1$ .

$- (12 x^{2} - 6 x) + 2 (12 {(2 x)}^{2} - 6 (2 x))$

Step 6.3.2

Factor $2$ out of $2 x$ .

$- (12 x^{2} - 6 x) + 2 (12 {(2 (x))}^{2} - 6 (2 x))$

Step 6.3.3

Simplify the expression.

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Step 6.3.3.1

Apply the product rule to $2 (x)$ .

$- (12 x^{2} - 6 x) + 2 (12 (2^{2} x^{2}) - 6 (2 x))$

Step 6.3.3.2

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

$- (12 x^{2} - 6 x) + 2 (12 (4 x^{2}) - 6 (2 x))$

Step 6.3.3.3

Multiply $4$ by $12$ .

$- (12 x^{2} - 6 x) + 2 (48 x^{2} - 6 (2 x))$

Step 6.3.3.4

Multiply $2$ by $- 6$ .

$- (12 x^{2} - 6 x) + 2 (48 x^{2} - 12 x)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$- (12 x^{2}) - (- 6 x) + 2 (48 x^{2} - 12 x)$

Step 7.2

Apply the distributive property.

$- (12 x^{2}) - (- 6 x) + 2 (48 x^{2}) + 2 (- 12 x)$

Step 7.3

Combine terms.

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Step 7.3.1

Multiply $12$ by $- 1$ .

$- 12 x^{2} - (- 6 x) + 2 (48 x^{2}) + 2 (- 12 x)$

Step 7.3.2

Multiply $- 6$ by $- 1$ .

$- 12 x^{2} + 6 x + 2 (48 x^{2}) + 2 (- 12 x)$

Step 7.3.3

Multiply $48$ by $2$ .

$- 12 x^{2} + 6 x + 96 x^{2} + 2 (- 12 x)$

Step 7.3.4

Multiply $- 12$ by $2$ .

$- 12 x^{2} + 6 x + 96 x^{2} - 24 x$

Step 7.3.5

Add $- 12 x^{2}$ and $96 x^{2}$ .

$84 x^{2} + 6 x - 24 x$

Step 7.3.6

Subtract $24 x$ from $6 x$ .

$84 x^{2} - 18 x$