Calculus Examples

$\int 16 x^{2} e^{4 x^{3}} d x$

Step 1

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 \int x^{2} e^{4 x^{3}} d x$

Step 2

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 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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 2.1.2.1

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 2.1.2.2

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 2.1.2.3

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

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

Step 2.1.3

Differentiate.

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Step 2.1.3.1

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 2.1.3.2

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 2.1.3.3

Multiply $3$ by $4$ .

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

Step 2.1.4

Simplify.

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Step 2.1.4.1

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

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

Step 2.1.4.2

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

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

Step 2.2

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

$16 \int \frac{1}{12} d u_{2}$

Step 3

Apply the constant rule.

$16 (\frac{1}{12} u_{2} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $16 (\frac{1}{12} u_{2} + C)$ as $16 (\frac{1}{12}) u_{2} + C$ .

$16 (\frac{1}{12}) u_{2} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $16$ and $\frac{1}{12}$ .

$\frac{16}{12} u_{2} + C$

Step 4.2.2

Cancel the common factor of $16$ and $12$ .

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Step 4.2.2.1

Factor $4$ out of $16$ .

$\frac{4 (4)}{12} u_{2} + C$

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 4}{4 \cdot 3} u_{2} + C$

Step 4.2.2.2.2

Cancel the common factor.

$\frac{4 \cdot 4}{4 \cdot 3} u_{2} + C$

Step 4.2.2.2.3

Rewrite the expression.

$\frac{4}{3} u_{2} + C$

Step 4.3

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

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