Calculus Examples

$\int_{0}^{\frac{π}{4}} \cos (2 x) e^{\sin (2 x)} d x$

Step 1

Let $u_{2} = \sin (2 x)$ . Then $d u_{2} = 2 \cos (2 x) d x$ , so $\frac{1}{2} d u_{2} = \cos (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \sin (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\sin (2 x)$ .

$\frac{d}{d x} [\sin (2 x)]$

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 $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 1.1.2.1

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

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [2 x]$

Step 1.1.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d x} [2 x]$

Step 1.1.2.3

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

$\cos (2 x) \frac{d}{d x} [2 x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$\cos (2 x) (2 \frac{d}{d x} [x])$

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 $n = 1$ .

$\cos (2 x) (2 \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $2$ by $1$ .

$\cos (2 x) \cdot 2$

Step 1.1.3.3.2

Move $2$ to the left of $\cos (2 x)$ .

$2 \cos (2 x)$

Step 1.2

Substitute the lower limit in for $x$ in $u_{2} = \sin (2 x)$ .

$u_{lower} = \sin (2 \cdot 0)$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $2$ by $0$ .

$u_{lower} = \sin (0)$

Step 1.3.2

The exact value of $\sin (0)$ is $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u_{2} = \sin (2 x)$ .

$u_{upper} = \sin (2 \frac{π}{4})$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $2$ .

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Step 1.5.1.1

Factor $2$ out of $4$ .

$u_{upper} = \sin (2 \frac{π}{2 (2)})$

Step 1.5.1.2

Cancel the common factor.

$u_{upper} = \sin (2 \frac{π}{2 \cdot 2})$

Step 1.5.1.3

Rewrite the expression.

$u_{upper} = \sin (\frac{π}{2})$

Step 1.5.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$u_{upper} = 1$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 1.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\int_{0}^{1} e^{u_{2}} \frac{1}{2} d u_{2}$

Step 2

Combine $e^{u_{2}}$ and $\frac{1}{2}$ .

$\int_{0}^{1} \frac{e^{u_{2}}}{2} d u_{2}$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{0}^{1} e^{u_{2}} d u_{2}$

Step 4

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$\frac{1}{2} e^{u_{2}}]_{0}^{1}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $e^{u_{2}}$ at $1$ and at $0$ .

$\frac{1}{2} ((e^{1}) - e^{0})$

Step 5.2

Simplify.

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Step 5.2.1

Simplify.

$\frac{1}{2} (e - e^{0})$

Step 5.2.2

Anything raised to $0$ is $1$ .

$\frac{1}{2} (e - 1 \cdot 1)$

Step 5.2.3

Multiply $- 1$ by $1$ .

$\frac{1}{2} (e - 1)$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{1}{2} e + \frac{1}{2} \cdot - 1$

Step 6.2

Combine $\frac{1}{2}$ and $e$ .

$\frac{e}{2} + \frac{1}{2} \cdot - 1$

Step 6.3

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{e}{2} + \frac{- 1}{2}$

Step 6.4

Move the negative in front of the fraction.

$\frac{e}{2} - \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{e}{2} - \frac{1}{2}$

Decimal Form:

$0.85914091 \dots$