Calculus Examples

$\int_{0}^{π} (6 + \sin (x)) d x$

Step 1

Remove parentheses.

$\int_{0}^{π} 6 + \sin (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{π} 6 d x + \int_{0}^{π} \sin (x) d x$

Step 3

Apply the constant rule.

$6 x]_{0}^{π} + \int_{0}^{π} \sin (x) d x$

Step 4

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

$6 x]_{0}^{π} + - \cos (x)]_{0}^{π}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Combine $6 x]_{0}^{π}$ and $- \cos (x)]_{0}^{π}$ .

$6 x - \cos (x)]_{0}^{π}$

Step 5.2

Substitute and simplify.

Tap for more steps...

Step 5.2.1

Evaluate $6 x - \cos (x)$ at $π$ and at $0$ .

$(6 π - \cos (π)) - (6 \cdot 0 - \cos (0))$

Step 5.2.2

Simplify.

Tap for more steps...

Step 5.2.2.1

Multiply $6$ by $0$ .

$6 π - \cos (π) - (0 - \cos (0))$

Step 5.2.2.2

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

$6 π - \cos (π) - - \cos (0)$

Step 5.2.2.3

Multiply $- 1$ by $- 1$ .

$6 π - \cos (π) + 1 \cos (0)$

Step 5.2.2.4

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

$6 π - \cos (π) + \cos (0)$

Step 5.3

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

$6 π - \cos (π) + 1$

Step 5.4

Simplify.

Tap for more steps...

Step 5.4.1

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.

$6 π - - \cos (0) + 1$

Step 5.4.2

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

$6 π - (- 1 \cdot 1) + 1$

Step 5.4.3

Multiply $- 1$ by $1$ .

$6 π - - 1 + 1$

Step 5.4.4

Multiply $- 1$ by $- 1$ .

$6 π + 1 + 1$

Step 5.4.5

Add $1$ and $1$ .

$6 π + 2$

Step 6

The result can be shown in multiple forms.

Exact Form:

$6 π + 2$

Decimal Form:

$20.84955592 \dots$