Calculus Examples

$\int_{0}^{7} \frac{3 y^{2} - 2 y + 5}{y^{3} - y^{2} + 5 y + 1} d y$

Step 1

Let $u = y^{3} - y^{2} + 5 y + 1$ . Then $d u = (3 y^{2} - 2 y + 5) d y$ , so $\frac{1}{3 y^{2} - 2 y + 5} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = y^{3} - y^{2} + 5 y + 1$ . Find $\frac{d u}{d y}$ .

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Step 1.1.1

Differentiate $y^{3} - y^{2} + 5 y + 1$ .

$\frac{d}{d y} [y^{3} - y^{2} + 5 y + 1]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $y^{3} - y^{2} + 5 y + 1$ with respect to $y$ is $\frac{d}{d y} [y^{3}] + \frac{d}{d y} [- y^{2}] + \frac{d}{d y} [5 y] + \frac{d}{d y} [1]$ .

$\frac{d}{d y} [y^{3}] + \frac{d}{d y} [- y^{2}] + \frac{d}{d y} [5 y] + \frac{d}{d y} [1]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 3$ .

$3 y^{2} + \frac{d}{d y} [- y^{2}] + \frac{d}{d y} [5 y] + \frac{d}{d y} [1]$

Step 1.1.3

Evaluate $\frac{d}{d y} [- y^{2}]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- y^{2}$ with respect to $y$ is $- \frac{d}{d y} [y^{2}]$ .

$3 y^{2} - \frac{d}{d y} [y^{2}] + \frac{d}{d y} [5 y] + \frac{d}{d y} [1]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$3 y^{2} - (2 y) + \frac{d}{d y} [5 y] + \frac{d}{d y} [1]$

Step 1.1.3.3

Multiply $2$ by $- 1$ .

$3 y^{2} - 2 y + \frac{d}{d y} [5 y] + \frac{d}{d y} [1]$

Step 1.1.4

Evaluate $\frac{d}{d y} [5 y]$ .

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Step 1.1.4.1

Since $5$ is constant with respect to $y$ , the derivative of $5 y$ with respect to $y$ is $5 \frac{d}{d y} [y]$ .

$3 y^{2} - 2 y + 5 \frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 1.1.4.2

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

$3 y^{2} - 2 y + 5 \cdot 1 + \frac{d}{d y} [1]$

Step 1.1.4.3

Multiply $5$ by $1$ .

$3 y^{2} - 2 y + 5 + \frac{d}{d y} [1]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

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

$3 y^{2} - 2 y + 5 + 0$

Step 1.1.5.2

Add $3 y^{2} - 2 y + 5$ and $0$ .

$3 y^{2} - 2 y + 5$

Step 1.2

Substitute the lower limit in for $y$ in $u = y^{3} - y^{2} + 5 y + 1$ .

$u_{lower} = 0^{3} - 0^{2} + 5 \cdot 0 + 1$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0 - 0^{2} + 5 \cdot 0 + 1$

Step 1.3.1.2

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0 - 0 + 5 \cdot 0 + 1$

Step 1.3.1.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0 + 0 + 5 \cdot 0 + 1$

Step 1.3.1.4

Multiply $5$ by $0$ .

$u_{lower} = 0 + 0 + 0 + 1$

Step 1.3.2

Add $0$ and $0$ .

$u_{lower} = 0 + 0 + 1$

Step 1.3.3

Add $0$ and $0$ .

$u_{lower} = 0 + 1$

Step 1.3.4

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $y$ in $u = y^{3} - y^{2} + 5 y + 1$ .

$u_{upper} = 7^{3} - 7^{2} + 5 \cdot 7 + 1$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $7$ to the power of $3$ .

$u_{upper} = 343 - 7^{2} + 5 \cdot 7 + 1$

Step 1.5.1.2

Raise $7$ to the power of $2$ .

$u_{upper} = 343 - 1 \cdot 49 + 5 \cdot 7 + 1$

Step 1.5.1.3

Multiply $- 1$ by $49$ .

$u_{upper} = 343 - 49 + 5 \cdot 7 + 1$

Step 1.5.1.4

Multiply $5$ by $7$ .

$u_{upper} = 343 - 49 + 35 + 1$

Step 1.5.2

Subtract $49$ from $343$ .

$u_{upper} = 294 + 35 + 1$

Step 1.5.3

Add $294$ and $35$ .

$u_{upper} = 329 + 1$

Step 1.5.4

Add $329$ and $1$ .

$u_{upper} = 330$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 330$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{330} \frac{1}{u} d u$

Step 2

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

$\ln (| u |)]_{1}^{330}$

Step 3

Evaluate $\ln (| u |)$ at $330$ and at $1$ .

$\ln (| 330 |) - \ln (| 1 |)$

Step 4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| 330 |}{| 1 |})$

Step 5

Simplify.

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Step 5.1

The absolute value is the distance between a number and zero. The distance between $0$ and $330$ is $330$ .

$\ln (\frac{330}{| 1 |})$

Step 5.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\ln (\frac{330}{1})$

Step 5.3

Divide $330$ by $1$ .

$\ln (330)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (330)$

Decimal Form:

$5.79909265 \dots$

Step 7