Calculus Examples

$\int 12 + \frac{500}{x + 1} d x$

Step 1

Split the single integral into multiple integrals.

$\int 12 d x + \int \frac{500}{x + 1} d x$

Step 2

Apply the constant rule.

$12 x + C + \int \frac{500}{x + 1} d x$

Step 3

Since $500$ is constant with respect to $x$ , move $500$ out of the integral.

$12 x + C + 500 \int \frac{1}{x + 1} d x$

Step 4

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 4.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 4.1.3

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

$1 + \frac{d}{d x} [1]$

Step 4.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

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

$12 x + C + 500 \int \frac{1}{u} d u$

Step 5

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$12 x + C + 500 (\ln (| u |) + C)$

Step 6

Simplify.

$12 x + 500 \ln (| u |) + C$

Step 7

Replace all occurrences of $u$ with $x + 1$ .

$12 x + 500 \ln (| x + 1 |) + C$