Calculus Examples

$\int_{0}^{π} x + \sin (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{π} x d x + \int_{0}^{π} \sin (x) d x$

Step 2

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{2} x^{2}]_{0}^{π}$ and $- \cos (x)]_{0}^{π}$ .

$\frac{1}{2} x^{2} - \cos (x)]_{0}^{π}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{1}{2} x^{2} - \cos (x)$ at $π$ and at $0$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Combine $\frac{1}{2}$ and $π^{2}$ .

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

Step 4.2.2.2

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

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

Step 4.2.2.3

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

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

Step 4.2.2.4

Subtract $\cos (0)$ from $0$ .

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

Step 4.2.2.5

Multiply $- 1$ by $- 1$ .

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

Step 4.2.2.6

Multiply $\cos (0)$ by $1$ .

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

Step 4.3

The exact value of $\cos (0)$ is $1$ .

$\frac{π^{2}}{2} - \cos (π) + 1$

Step 4.4

Simplify.

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Step 4.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{π^{2}}{2} - - \cos (0) + 1$

Step 4.4.2

The exact value of $\cos (0)$ is $1$ .

$\frac{π^{2}}{2} - (- 1 \cdot 1) + 1$

Step 4.4.3

Multiply $- 1$ by $1$ .

$\frac{π^{2}}{2} - - 1 + 1$

Step 4.4.4

Multiply $- 1$ by $- 1$ .

$\frac{π^{2}}{2} + 1 + 1$

Step 4.4.5

Add $1$ and $1$ .

$\frac{π^{2}}{2} + 2$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{π^{2}}{2} + 2$

Decimal Form:

$6.93480220 \dots$