Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify.

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Step 4.3.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

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

Step 4.3.2

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

$- 0 + 1 - (- \cos (0) + \sin (0))$

Step 4.3.3

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

$- 0 + 1 - (- 1 \cdot 1 + \sin (0))$

Step 4.3.4

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

$- 0 + 1 - (- 1 \cdot 1 + 0)$

Step 4.3.5

Multiply $- 1$ by $0$ .

$0 + 1 - (- 1 \cdot 1 + 0)$

Step 4.3.6

Add $0$ and $1$ .

$1 - (- 1 \cdot 1 + 0)$

Step 4.3.7

Multiply $- 1$ by $1$ .

$1 - (- 1 + 0)$

Step 4.3.8

Add $- 1$ and $0$ .

$1 - - 1$

Step 4.3.9

Multiply $- 1$ by $- 1$ .

$1 + 1$

Step 4.3.10

Add $1$ and $1$ .

$2$