Calculus Examples

$\int x^{2} e^{4 x^{3}} d x$

Step 1

Let $u_{2} = e^{4 x^{3}}$ . Then $d u_{2} = 12 x^{2} e^{4 x^{3}} d x$ , so $\frac{1}{12} d u_{2} = x^{2} e^{4 x^{3}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = e^{4 x^{3}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $e^{4 x^{3}}$ .

$\frac{d}{d x} [e^{4 x^{3}}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 4 x^{3}$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $4 x^{3}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [4 x^{3}]$

Step 1.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [4 x^{3}]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $4 x^{3}$ .

$e^{4 x^{3}} \frac{d}{d x} [4 x^{3}]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$e^{4 x^{3}} (4 \frac{d}{d x} [x^{3}])$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$e^{4 x^{3}} (4 (3 x^{2}))$

Step 1.1.3.3

Multiply $3$ by $4$ .

$e^{4 x^{3}} (12 x^{2})$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Reorder the factors of $e^{4 x^{3}} (12 x^{2})$ .

$12 e^{4 x^{3}} x^{2}$

Step 1.1.4.2

Reorder factors in $12 e^{4 x^{3}} x^{2}$ .

$12 x^{2} e^{4 x^{3}}$

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \frac{1}{12} d u_{2}$

Step 2

Apply the constant rule.

$\frac{1}{12} u_{2} + C$

Step 3

Replace all occurrences of $u_{2}$ with $e^{4 x^{3}}$ .

$\frac{1}{12} e^{4 x^{3}} + C$