Calculus Examples

$\int_{x}^{x^{2}} e^{t^{7}} d t$

Step 1

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

$\frac{d}{d x} [\int_{x}^{c} e^{t^{7}} d t + \int_{c}^{x^{2}} e^{t^{7}} d t]$

Step 2

By the Sum Rule, the derivative of $\int_{x}^{c} e^{t^{7}} d t + \int_{c}^{x^{2}} e^{t^{7}} d t$ with respect to $x$ is $\frac{d}{d x} [\int_{x}^{c} e^{t^{7}} d t] + \frac{d}{d x} [\int_{c}^{x^{2}} e^{t^{7}} d t]$ .

$\frac{d}{d x} [\int_{x}^{c} e^{t^{7}} d t] + \frac{d}{d x} [\int_{c}^{x^{2}} e^{t^{7}} d t]$

Step 3

Swap the bounds of integration.

$\frac{d}{d x} [- \int_{c}^{x} e^{t^{7}} d t] + \frac{d}{d x} [\int_{c}^{x^{2}} e^{t^{7}} d t]$

Step 4

Take the derivative of $- \int_{c}^{x} e^{t^{7}} d t$ with respect to $x$ using Fundamental Theorem of Calculus.

$- e^{x^{7}} + \frac{d}{d x} [\int_{c}^{x^{2}} e^{t^{7}} d t]$

Step 5

Take the derivative of $\int_{c}^{x^{2}} e^{t^{7}} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$- e^{x^{7}} + \frac{d}{d x} [x^{2}] e^{{(x^{2})}^{7}}$

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 $n = 2$ .

$- e^{x^{7}} + 2 x e^{{(x^{2})}^{7}}$

Step 6.2

Multiply the exponents in ${(x^{2})}^{7}$ .

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Step 6.2.1

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

$- e^{x^{7}} + 2 x e^{x^{2 \cdot 7}}$

Step 6.2.2

Multiply $2$ by $7$ .

$- e^{x^{7}} + 2 x e^{x^{14}}$

Step 7

Simplify.

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Step 7.1

Reorder terms.

$2 e^{x^{14}} x - e^{x^{7}}$

Step 7.2

Reorder factors in $2 e^{x^{14}} x - e^{x^{7}}$ .

$2 x e^{x^{14}} - e^{x^{7}}$