Calculus Examples

$\int_{\frac{π}{2}}^{π} e^{\sin (q)} \cdot \cos (q) d q$

Step 1

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

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Step 1.1

Let $u = \sin (q)$ . Find $\frac{d u}{d q}$ .

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Step 1.1.1

Differentiate $\sin (q)$ .

$\frac{d}{d q} [\sin (q)]$

Step 1.1.2

The derivative of $\sin (q)$ with respect to $q$ is $\cos (q)$ .

$\cos (q)$

Step 1.2

Substitute the lower limit in for $q$ in $u = \sin (q)$ .

$u_{lower} = \sin (\frac{π}{2})$

Step 1.3

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

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $q$ in $u = \sin (q)$ .

$u_{upper} = \sin (π)$

Step 1.5

Simplify.

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Step 1.5.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 1.5.2

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

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 1.7

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

$\int_{1}^{0} e^{u} d u$

Step 2

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

$e^{u}]_{1}^{0}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $e^{u}$ at $0$ and at $1$ .

$(e^{0}) - e^{1}$

Step 3.2

Simplify.

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Step 3.2.1

Anything raised to $0$ is $1$ .

$1 - e^{1}$

Step 3.2.2

Simplify.

$1 - e$

Step 4

The result can be shown in multiple forms.

Exact Form:

$1 - e$

Decimal Form:

$- 1.71828182 \dots$