Calculus Examples

$\int 3 x e^{0.5 x^{2}} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x e^{0.5 x^{2}} d x$

Step 2

Let $u_{2} = e^{0.5 x^{2}}$ . Then $d u_{2} = x e^{0.5 x^{2}} d x$ , so $\frac{1}{x e^{0.5 x^{2}}} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = e^{0.5 x^{2}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.1.1

Differentiate $e^{0.5 x^{2}}$ .

$\frac{d}{d x} [e^{0.5 x^{2}}]$

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 $g (x) = 0.5 x^{2}$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $0.5 x^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [0.5 x^{2}]$

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 $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [0.5 x^{2}]$

Step 2.1.2.3

Replace all occurrences of $u_{1}$ with $0.5 x^{2}$ .

$e^{0.5 x^{2}} \frac{d}{d x} [0.5 x^{2}]$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

Since $0.5$ is constant with respect to $x$ , the derivative of $0.5 x^{2}$ with respect to $x$ is $0.5 \frac{d}{d x} [x^{2}]$ .

$e^{0.5 x^{2}} (0.5 \frac{d}{d x} [x^{2}])$

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 $n = 2$ .

$e^{0.5 x^{2}} (0.5 (2 x))$

Step 2.1.3.3

Simplify the expression.

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Step 2.1.3.3.1

Multiply $2$ by $0.5$ .

$e^{0.5 x^{2}} (1 x)$

Step 2.1.3.3.2

Multiply $x$ by $1$ .

$e^{0.5 x^{2}} x$

Step 2.1.3.3.3

Reorder factors in $e^{0.5 x^{2}} x$ .

$x e^{0.5 x^{2}}$

Step 2.2

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

$3 \int d u_{2}$

Step 3

Apply the constant rule.

$3 (u_{2} + C)$

Step 4

Simplify the answer.

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Step 4.1

Simplify.

$3 u_{2} + C$

Step 4.2

Replace all occurrences of $u_{2}$ with $e^{0.5 x^{2}}$ .

$3 e^{0.5 x^{2}} + C$