Calculus Examples

$\int_{x}^{x^{2}} t^{2} d t$

Step 1

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

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

Step 2

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

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

Step 3

Swap the bounds of integration.

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

Step 4

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

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

Step 5

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

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

Step 6

Differentiate using the Power Rule.

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Step 6.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $2$ .

$- x^{2} + 2 x \cdot x^{4}$

Step 7

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 7.1

Move $x^{4}$ .

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

Step 7.2

Multiply $x^{4}$ by $x$ .

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Step 7.2.1

Raise $x$ to the power of $1$ .

$- x^{2} + 2 (x^{4} x^{1})$

Step 7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{2} + 2 x^{4 + 1}$

Step 7.3

Add $4$ and $1$ .

$- x^{2} + 2 x^{5}$

Step 8

Reorder terms.

$2 x^{5} - x^{2}$