Calculus Examples

$\int_{0}^{\frac{π}{2}} x d x$

Step 1

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{2} x^{2}]_{0}^{\frac{π}{2}}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{2} x^{2}$ at $\frac{π}{2}$ and at $0$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Raising $0$ to any positive power yields $0$ .

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

Step 2.2.2

Multiply $0$ by $- 1$ .

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

Step 2.2.3

Multiply $0$ by $\frac{1}{2}$ .

$\frac{1}{2} {(\frac{π}{2})}^{2} + 0$

Step 2.2.4

Add $\frac{1}{2} {(\frac{π}{2})}^{2}$ and $0$ .

$\frac{1}{2} {(\frac{π}{2})}^{2}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{π}{2}$ .

$\frac{1}{2} \cdot \frac{π^{2}}{2^{2}}$

Step 3.2

Combine.

$\frac{1 π^{2}}{2 \cdot 2^{2}}$

Step 3.3

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 3.3.1

Multiply $2$ by $2^{2}$ .

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Step 3.3.1.1

Raise $2$ to the power of $1$ .

$\frac{1 π^{2}}{2^{1} \cdot 2^{2}}$

Step 3.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1 π^{2}}{2^{1 + 2}}$

Step 3.3.2

Add $1$ and $2$ .

$\frac{1 π^{2}}{2^{3}}$

Step 3.4

Multiply $π^{2}$ by $1$ .

$\frac{π^{2}}{2^{3}}$

Step 3.5

Raise $2$ to the power of $3$ .

$\frac{π^{2}}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{π^{2}}{8}$

Decimal Form:

$1.23370055 \dots$

Step 5