Calculus Examples

$\int_{- x}^{x} (t^{2} + t) d t$

Step 1

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

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

Step 2

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

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

Step 3

Swap the bounds of integration.

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

Step 4

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

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 6

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

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

Step 7

Simplify terms.

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Step 7.1

Factor $- 1$ out of $- x$ .

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

Step 7.2

Simplify the expression.

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Step 7.2.1

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

$- - ({(- 1)}^{2} x^{2} - x) + x^{2} + x$

Step 7.2.2

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

$- - (1 x^{2} - x) + x^{2} + x$

Step 7.2.3

Multiply $x^{2}$ by $1$ .

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

Step 7.2.4

Multiply $- 1$ by $- 1$ .

$1 (x^{2} - x) + x^{2} + x$

Step 7.2.5

Multiply $x^{2} - x$ by $1$ .

$x^{2} - x + x^{2} + x$

Step 7.3

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} - x + x$

Step 7.4

Add $- x$ and $x$ .

$2 x^{2} + 0$

Step 7.5

Add $2 x^{2}$ and $0$ .

$2 x^{2}$