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

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 4

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 5

Simplify the answer.

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Step 5.1

Substitute and simplify.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

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

$- 0 + 1 - (1 - 0)$

Step 5.2.5

Multiply $- 1$ by $0$ .

$0 + 1 - (1 - 0)$

Step 5.2.6

Add $0$ and $1$ .

$1 - (1 - 0)$

Step 5.2.7

Multiply $- 1$ by $0$ .

$1 - (1 + 0)$

Step 5.2.8

Add $1$ and $0$ .

$1 - 1 \cdot 1$

Step 5.2.9

Multiply $- 1$ by $1$ .

$1 - 1$

Step 5.2.10

Subtract $1$ from $1$ .

$0$