Calculus Examples

$\int (x + 3) {(x - 1)}^{5} d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 1$ .

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

Step 1.1.2

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 1.1.3

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 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$\int (u + 1 + 3) u^{5} d u$

Step 2

Add $1$ and $3$ .

$\int (u + 4) u^{5} d u$

Step 3

Multiply $(u + 4) u^{5}$ .

$\int u \cdot u^{5} + 4 u^{5} d u$

Step 4

Multiply $u$ by $u^{5}$ by adding the exponents.

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Step 4.1

Multiply $u$ by $u^{5}$ .

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Step 4.1.1

Raise $u$ to the power of $1$ .

$\int u^{1} u^{5} + 4 u^{5} d u$

Step 4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{1 + 5} + 4 u^{5} d u$

Step 4.2

Add $1$ and $5$ .

$\int u^{6} + 4 u^{5} d u$

Step 5

Split the single integral into multiple integrals.

$\int u^{6} d u + \int 4 u^{5} d u$

Step 6

By the Power Rule, the integral of $u^{6}$ with respect to $u$ is $\frac{1}{7} u^{7}$ .

$\frac{1}{7} u^{7} + C + \int 4 u^{5} d u$

Step 7

Since $4$ is constant with respect to $u$ , move $4$ out of the integral.

$\frac{1}{7} u^{7} + C + 4 \int u^{5} d u$

Step 8

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{7} u^{7} + C + 4 (\frac{1}{6} u^{6} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{1}{7} u^{7} + 4 (\frac{1}{6}) u^{6} + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $4$ and $\frac{1}{6}$ .

$\frac{1}{7} u^{7} + \frac{4}{6} u^{6} + C$

Step 9.2.2

Cancel the common factor of $4$ and $6$ .

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Step 9.2.2.1

Factor $2$ out of $4$ .

$\frac{1}{7} u^{7} + \frac{2 (2)}{6} u^{6} + C$

Step 9.2.2.2

Cancel the common factors.

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Step 9.2.2.2.1

Factor $2$ out of $6$ .

$\frac{1}{7} u^{7} + \frac{2 \cdot 2}{2 \cdot 3} u^{6} + C$

Step 9.2.2.2.2

Cancel the common factor.

$\frac{1}{7} u^{7} + \frac{2 \cdot 2}{2 \cdot 3} u^{6} + C$

Step 9.2.2.2.3

Rewrite the expression.

$\frac{1}{7} u^{7} + \frac{2}{3} u^{6} + C$

Step 10

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

$\frac{1}{7} {(x - 1)}^{7} + \frac{2}{3} {(x - 1)}^{6} + C$