Calculus Examples

$f (x) = 7 x {(x - 1)}^{6}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int 7 x {(x - 1)}^{6} d x$

Step 3

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int x {(x - 1)}^{6} d x$

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 $d u$ .

$7 \int (u + 1) u^{6} d u$

Step 5

Multiply $(u + 1) u^{6}$ .

$7 \int u \cdot u^{6} + 1 u^{6} d u$

Step 6

Simplify.

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Step 6.1

Multiply $u$ by $u^{6}$ by adding the exponents.

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Step 6.1.1

Multiply $u$ by $u^{6}$ .

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Step 6.1.1.1

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

$7 \int u^{1} u^{6} + 1 u^{6} d u$

Step 6.1.1.2

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

$7 \int u^{1 + 6} + 1 u^{6} d u$

Step 6.1.2

Add $1$ and $6$ .

$7 \int u^{7} + 1 u^{6} d u$

Step 6.2

Multiply $u^{6}$ by $1$ .

$7 \int u^{7} + u^{6} d u$

Step 7

Split the single integral into multiple integrals.

$7 (\int u^{7} d u + \int u^{6} d u)$

Step 8

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$7 (\frac{1}{8} u^{8} + C + \int u^{6} d u)$

Step 9

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

$7 (\frac{1}{8} u^{8} + C + \frac{1}{7} u^{7} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{8}$ and $u^{8}$ .

$7 (\frac{u^{8}}{8} + C + \frac{1}{7} u^{7} + C)$

Step 10.1.2

Combine $\frac{1}{7}$ and $u^{7}$ .

$7 (\frac{u^{8}}{8} + C + \frac{u^{7}}{7} + C)$

Step 10.2

Simplify.

$7 (\frac{1}{8} u^{8} + \frac{1}{7} u^{7}) + C$

Step 11

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

$7 (\frac{1}{8} {(x - 1)}^{8} + \frac{1}{7} {(x - 1)}^{7}) + C$

Step 12

The answer is the antiderivative of the function $f (x) = 7 x {(x - 1)}^{6}$ .

$F (x) =$ $7 (\frac{1}{8} {(x - 1)}^{8} + \frac{1}{7} {(x - 1)}^{7}) + C$