Calculus Examples

$\int {(x - 3)}^{2} + 3 d x$

Step 1

Split the single integral into multiple integrals.

$\int {(x - 3)}^{2} d x + \int 3 d x$

Step 2

Let $u = x - 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x - 3$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x - 3$ .

$\frac{d}{d x} [x - 3]$

Step 2.1.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 2.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 2.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

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

$\int u^{2} d u + \int 3 d x$

Step 3

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{3} u^{3} + C + \int 3 d x$

Step 4

Apply the constant rule.

$\frac{1}{3} u^{3} + C + 3 x + C$

Step 5

Simplify.

$\frac{1}{3} u^{3} + 3 x + C$

Step 6

Replace all occurrences of $u$ with $x - 3$ .

$\frac{1}{3} {(x - 3)}^{3} + 3 x + C$