Calculus Examples

$\int 1 - \sin (π x) d x$

Split the single integral into multiple integrals.

$\int d x + \int - \sin (π x) d x$

Apply the constant rule.

$x + C + \int - \sin (π x) d x$

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

$x + C - \int \sin (π x) d x$

Let $u = π x$ . Then $d u = π d x$ , so $\frac{1}{π} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Let $u = π x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Since $π$ is constant with respect to $x$ , the derivative of $π x$ with respect to $x$ is $π \frac{d}{d x} [x]$ .

$π \frac{d}{d x} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$π \cdot 1$

Multiply $π$ by $1$ .

$π$

Rewrite the problem using $u$ and $d u$ .

$x + C - \int \sin (u) \frac{1}{π} d u$

Combine $\sin (u)$ and $\frac{1}{π}$ .

$x + C - \int \frac{\sin (u)}{π} d u$

Since $\frac{1}{π}$ is constant with respect to $u$ , move $\frac{1}{π}$ out of the integral.

$x + C - (\frac{1}{π} \int \sin (u) d u)$

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

$x + C - \frac{1}{π} (- \cos (u) + C)$

Simplify.

$x + \frac{\cos (u)}{π} + C$

Replace all occurrences of $u$ with $π x$ .

$x + \frac{\cos (π x)}{π} + C$

Reorder terms.

$x + \frac{1}{π} \cos (π x) + C$