Calculus Examples

$\int_{1}^{2} 2 e^{- 4 x} - \frac{1}{x^{2}} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{2} 2 e^{- 4 x} d x + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{1}^{2} e^{- 4 x} d x + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 3

Let $u = - 4 x$ . Then $d u = - 4 d x$ , so $- \frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = - 4 x$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $- 4 x$ .

$\frac{d}{d x} [- 4 x]$

Step 3.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x]$

Step 3.1.3

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

$- 4 \cdot 1$

Step 3.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 3.2

Substitute the lower limit in for $x$ in $u = - 4 x$ .

$u_{lower} = - 4 \cdot 1$

Step 3.3

Multiply $- 4$ by $1$ .

$u_{lower} = - 4$

Step 3.4

Substitute the upper limit in for $x$ in $u = - 4 x$ .

$u_{upper} = - 4 \cdot 2$

Step 3.5

Multiply $- 4$ by $2$ .

$u_{upper} = - 8$

Step 3.6

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

$u_{lower} = - 4$

$u_{upper} = - 8$

Step 3.7

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

$2 \int_{- 4}^{- 8} e^{u} \frac{1}{- 4} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

$2 \int_{- 4}^{- 8} e^{u} (- \frac{1}{4}) d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 4.2

Combine $e^{u}$ and $\frac{1}{4}$ .

$2 \int_{- 4}^{- 8} - \frac{e^{u}}{4} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$2 (- \int_{- 4}^{- 8} \frac{e^{u}}{4} d u) + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 6

Multiply $- 1$ by $2$ .

$- 2 \int_{- 4}^{- 8} \frac{e^{u}}{4} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 7

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$- 2 (\frac{1}{4} \int_{- 4}^{- 8} e^{u} d u) + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{4}$ and $- 2$ .

$\frac{- 2}{4} \int_{- 4}^{- 8} e^{u} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 8.2

Cancel the common factor of $- 2$ and $4$ .

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Step 8.2.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{4} \int_{- 4}^{- 8} e^{u} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot - 1}{2 \cdot 2} \int_{- 4}^{- 8} e^{u} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 8.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 2} \int_{- 4}^{- 8} e^{u} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 8.2.2.3

Rewrite the expression.

$\frac{- 1}{2} \int_{- 4}^{- 8} e^{u} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 8.3

Move the negative in front of the fraction.

$- \frac{1}{2} \int_{- 4}^{- 8} e^{u} d u + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 9

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- \frac{1}{2} e^{u}]_{- 4}^{- 8} + \int_{1}^{2} - \frac{1}{x^{2}} d x$

Step 10

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \frac{1}{2} e^{u}]_{- 4}^{- 8} - \int_{1}^{2} \frac{1}{x^{2}} d x$

Step 11

Apply basic rules of exponents.

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Step 11.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{2} e^{u}]_{- 4}^{- 8} - \int_{1}^{2} {(x^{2})}^{- 1} d x$

Step 11.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 11.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{2} e^{u}]_{- 4}^{- 8} - \int_{1}^{2} x^{2 \cdot - 1} d x$

Step 11.2.2

Multiply $2$ by $- 1$ .

$- \frac{1}{2} e^{u}]_{- 4}^{- 8} - \int_{1}^{2} x^{- 2} d x$

Step 12

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$- \frac{1}{2} e^{u}]_{- 4}^{- 8} - (- x^{- 1}]_{1}^{2})$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $e^{u}$ at $- 8$ and at $- 4$ .

$- \frac{1}{2} ((e^{- 8}) - e^{- 4}) - (- x^{- 1}]_{1}^{2})$

Step 13.2

Evaluate $- x^{- 1}$ at $2$ and at $1$ .

$- \frac{1}{2} ((e^{- 8}) - e^{- 4}) - ((- 2^{- 1}) + 1^{- 1})$

Step 13.3

Simplify.

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Step 13.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{2} (e^{- 8} - e^{- 4}) - (- \frac{1}{2} + 1^{- 1})$

Step 13.3.2

One to any power is one.

$- \frac{1}{2} (e^{- 8} - e^{- 4}) - (- \frac{1}{2} + 1)$

Step 13.3.3

Write $1$ as a fraction with a common denominator.

$- \frac{1}{2} (e^{- 8} - e^{- 4}) - (- \frac{1}{2} + \frac{2}{2})$

Step 13.3.4

Combine the numerators over the common denominator.

$- \frac{1}{2} (e^{- 8} - e^{- 4}) - \frac{- 1 + 2}{2}$

Step 13.3.5

Add $- 1$ and $2$ .

$- \frac{1}{2} (e^{- 8} - e^{- 4}) - \frac{1}{2}$

Step 13.3.6

To write $- \frac{1}{2} (e^{- 8} - e^{- 4})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{2} (e^{- 8} - e^{- 4}) \cdot \frac{2}{2} - \frac{1}{2}$

Step 13.3.7

Combine $- \frac{1}{2} (e^{- 8} - e^{- 4})$ and $\frac{2}{2}$ .

$\frac{- \frac{1}{2} (e^{- 8} - e^{- 4}) \cdot 2}{2} - \frac{1}{2}$

Step 13.3.8

Combine the numerators over the common denominator.

$\frac{- \frac{1}{2} (e^{- 8} - e^{- 4}) \cdot 2 - 1}{2}$

Step 13.3.9

Multiply $2$ by $- 1$ .

$\frac{- 2 (\frac{1}{2}) (e^{- 8} - e^{- 4}) - 1}{2}$

Step 13.3.10

Combine $- 2$ and $\frac{1}{2}$ .

$\frac{\frac{- 2}{2} (e^{- 8} - e^{- 4}) - 1}{2}$

Step 13.3.11

Cancel the common factor of $- 2$ and $2$ .

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Step 13.3.11.1

Factor $2$ out of $- 2$ .

$\frac{\frac{2 \cdot - 1}{2} (e^{- 8} - e^{- 4}) - 1}{2}$

Step 13.3.11.2

Cancel the common factors.

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Step 13.3.11.2.1

Factor $2$ out of $2$ .

$\frac{\frac{2 \cdot - 1}{2 (1)} (e^{- 8} - e^{- 4}) - 1}{2}$

Step 13.3.11.2.2

Cancel the common factor.

$\frac{\frac{2 \cdot - 1}{2 \cdot 1} (e^{- 8} - e^{- 4}) - 1}{2}$

Step 13.3.11.2.3

Rewrite the expression.

$\frac{\frac{- 1}{1} (e^{- 8} - e^{- 4}) - 1}{2}$

Step 13.3.11.2.4

Divide $- 1$ by $1$ .

$\frac{- (e^{- 8} - e^{- 4}) - 1}{2}$

Step 14

Simplify.

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Step 14.1

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- (e^{- 8} - e^{- 4}) - 1 (1)}{2}$

Step 14.2

Factor $- 1$ out of $- (e^{- 8} - e^{- 4}) - 1 (1)$ .

$\frac{- (e^{- 8} - e^{- 4} + 1)}{2}$

Step 14.3

Rewrite $- (e^{- 8} - e^{- 4} + 1)$ as $- 1 (e^{- 8} - e^{- 4} + 1)$ .

$\frac{- 1 (e^{- 8} - e^{- 4} + 1)}{2}$

Step 14.4

Move the negative in front of the fraction.

$- \frac{e^{- 8} - e^{- 4} + 1}{2}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{e^{- 8} - e^{- 4} + 1}{2}$

Decimal Form:

$- 0.49100991 \dots$

Step 16