Calculus Examples

$\int_{\frac{π}{2}}^{0} \frac{1 + \cos (2 t)}{2} d t$

Step 1

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

$\frac{1}{2} \int_{\frac{π}{2}}^{0} 1 + \cos (2 t) d t$

Step 2

Split the single integral into multiple integrals.

$\frac{1}{2} (\int_{\frac{π}{2}}^{0} d t + \int_{\frac{π}{2}}^{0} \cos (2 t) d t)$

Step 3

Apply the constant rule.

$\frac{1}{2} (t]_{\frac{π}{2}}^{0} + \int_{\frac{π}{2}}^{0} \cos (2 t) d t)$

Step 4

Let $u = 2 t$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 4.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 4.1.2

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

$2 \frac{d}{d t} [t]$

Step 4.1.3

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

$2 \cdot 1$

Step 4.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2

Substitute the lower limit in for $t$ in $u = 2 t$ .

$u_{lower} = 2 \frac{π}{2}$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

$u_{lower} = 2 \frac{π}{2}$

Step 4.3.2

Rewrite the expression.

$u_{lower} = π$

Step 4.4

Substitute the upper limit in for $t$ in $u = 2 t$ .

$u_{upper} = 2 \cdot 0$

Step 4.5

Multiply $2$ by $0$ .

$u_{upper} = 0$

Step 4.6

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

$u_{lower} = π$

$u_{upper} = 0$

Step 4.7

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

$\frac{1}{2} (t]_{\frac{π}{2}}^{0} + \int_{π}^{0} \cos (u) \frac{1}{2} d u)$

Step 5

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{1}{2} (t]_{\frac{π}{2}}^{0} + \int_{π}^{0} \frac{\cos (u)}{2} d u)$

Step 6

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

$\frac{1}{2} (t]_{\frac{π}{2}}^{0} + \frac{1}{2} \int_{π}^{0} \cos (u) d u)$

Step 7

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

$\frac{1}{2} (t]_{\frac{π}{2}}^{0} + \frac{1}{2} \sin (u)]_{π}^{0})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $t$ at $0$ and at $\frac{π}{2}$ .

$\frac{1}{2} ((0) - \frac{π}{2} + \frac{1}{2} \sin (u)]_{π}^{0})$

Step 8.2

Evaluate $\sin (u)$ at $0$ and at $π$ .

$\frac{1}{2} ((0) - \frac{π}{2} + \frac{1}{2} ((\sin (0)) - \sin (π)))$

Step 8.3

Subtract $\frac{π}{2}$ from $0$ .

$\frac{1}{2} (- \frac{π}{2} + \frac{1}{2} (\sin (0) - \sin (π)))$

Step 9

Simplify.

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Step 9.1

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

$\frac{1}{2} (- \frac{π}{2} + \frac{1}{2} (0 - \sin (π)))$

Step 9.2

Subtract $\sin (π)$ from $0$ .

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

Step 9.3

Combine $\frac{1}{2}$ and $\sin (π)$ .

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

Step 10

Simplify.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$\frac{1}{2} (- \frac{π}{2} - \frac{\sin (0)}{2})$

Step 10.1.2

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

$\frac{1}{2} (- \frac{π}{2} - \frac{0}{2})$

Step 10.2

Divide $0$ by $2$ .

$\frac{1}{2} (- \frac{π}{2} - 0)$

Step 10.3

Multiply $- 1$ by $0$ .

$\frac{1}{2} (- \frac{π}{2} + 0)$

Step 10.4

Add $- \frac{π}{2}$ and $0$ .

$\frac{1}{2} (- \frac{π}{2})$

Step 10.5

Multiply $\frac{1}{2} (- \frac{π}{2})$ .

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Step 10.5.1

Multiply $\frac{1}{2}$ by $\frac{π}{2}$ .

$- \frac{π}{2 \cdot 2}$

Step 10.5.2

Multiply $2$ by $2$ .

$- \frac{π}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{π}{4}$

Decimal Form:

$- 0.78539816 \dots$