Calculus Examples

$\int_{0}^{\frac{π}{4}} \sin (t) + \cos (t) d t$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{\frac{π}{4}} \sin (t) d t + \int_{0}^{\frac{π}{4}} \cos (t) d t$

Step 2

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

$- \cos (t)]_{0}^{\frac{π}{4}} + \int_{0}^{\frac{π}{4}} \cos (t) d t$

Step 3

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

$- \cos (t)]_{0}^{\frac{π}{4}} + \sin (t)]_{0}^{\frac{π}{4}}$

Step 4

Simplify the answer.

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Step 4.1

Combine $- \cos (t)]_{0}^{\frac{π}{4}}$ and $\sin (t)]_{0}^{\frac{π}{4}}$ .

$- \cos (t) + \sin (t)]_{0}^{\frac{π}{4}}$

Step 4.2

Evaluate $- \cos (t) + \sin (t)$ at $\frac{π}{4}$ and at $0$ .

$- \cos (\frac{π}{4}) + \sin (\frac{π}{4}) - (- \cos (0) + \sin (0))$

Step 4.3

Simplify.

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Step 4.3.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} + \sin (\frac{π}{4}) - (- \cos (0) + \sin (0))$

Step 4.3.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 4.3.3

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

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

Step 4.3.4

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

$- \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - (- 1 \cdot 1 + 0)$

Step 4.3.5

Add $- \frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$ .

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

Step 4.3.6

Multiply $- 1$ by $1$ .

$0 - (- 1 + 0)$

Step 4.3.7

Add $- 1$ and $0$ .

$0 - - 1$

Step 4.3.8

Multiply $- 1$ by $- 1$ .

$0 + 1$

Step 4.3.9

Add $0$ and $1$ .

$1$