Calculus Examples

$\int_{- (\frac{5 π}{2})}^{\frac{5 π}{2}} \cos (x - 1) d x$

Step 1

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.1.3

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

$1 + \frac{d}{d x} [- 1]$

Step 1.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Substitute the lower limit in for $x$ in $u = x - 1$ .

$u_{lower} = - (\frac{5 π}{2}) - 1$

Step 1.3

Substitute the upper limit in for $x$ in $u = x - 1$ .

$u_{upper} = \frac{5 π}{2} - 1$

Step 1.4

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

$u_{lower} = - \frac{5 π}{2} - 1$

$u_{upper} = \frac{5 π}{2} - 1$

Step 1.5

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{- \frac{5 π}{2} - 1}^{\frac{5 π}{2} - 1} \cos (u) d u$

Step 2

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\sin (u)]_{- \frac{5 π}{2} - 1}^{\frac{5 π}{2} - 1}$

Step 3

Evaluate $\sin (u)$ at $\frac{5 π}{2} - 1$ and at $- \frac{5 π}{2} - 1$ .

$\sin (\frac{5 π}{2} - 1) - \sin (- \frac{5 π}{2} - 1)$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Evaluate $\sin (\frac{5 π}{2} - 1)$ .

$0.5403023 - \sin (- \frac{5 π}{2} - 1)$

Step 4.1.2

Evaluate $\sin (- \frac{5 π}{2} - 1)$ .

$0.5403023 - - 0.5403023$

Step 4.1.3

Multiply $- 1$ by $- 0.5403023$ .

$0.5403023 + 0.5403023$

Step 4.2

Add $0.5403023$ and $0.5403023$ .

$1.08060461$