Calculus Examples

$\int_{- 3}^{4} 3 e^{x} - 4 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 3}^{4} 3 e^{x} d x + \int_{- 3}^{4} - 4 d x$

Step 2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{- 3}^{4} e^{x} d x + \int_{- 3}^{4} - 4 d x$

Step 3

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

$3 (e^{x}]_{- 3}^{4}) + \int_{- 3}^{4} - 4 d x$

Step 4

Apply the constant rule.

$3 (e^{x}]_{- 3}^{4}) + - 4 x]_{- 3}^{4}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $e^{x}$ at $4$ and at $- 3$ .

$3 ((e^{4}) - e^{- 3}) + - 4 x]_{- 3}^{4}$

Step 5.2

Evaluate $- 4 x$ at $4$ and at $- 3$ .

$3 (e^{4} - e^{- 3}) + - 4 \cdot 4 + 4 \cdot - 3$

Step 5.3

Simplify.

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Step 5.3.1

Multiply $- 4$ by $4$ .

$3 (e^{4} - e^{- 3}) - 16 + 4 \cdot - 3$

Step 5.3.2

Multiply $4$ by $- 3$ .

$3 (e^{4} - e^{- 3}) - 16 - 12$

Step 5.3.3

Subtract $12$ from $- 16$ .

$3 (e^{4} - e^{- 3}) - 28$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$3 e^{4} + 3 (- e^{- 3}) - 28$

Step 6.1.2

Multiply $- 1$ by $3$ .

$3 e^{4} - 3 e^{- 3} - 28$

Step 6.2

Simplify each term.

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Step 6.2.1

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

$3 e^{4} - 3 \frac{1}{e^{3}} - 28$

Step 6.2.2

Combine $- 3$ and $\frac{1}{e^{3}}$ .

$3 e^{4} + \frac{- 3}{e^{3}} - 28$

Step 6.2.3

Move the negative in front of the fraction.

$3 e^{4} - \frac{3}{e^{3}} - 28$

Step 7

The result can be shown in multiple forms.

Exact Form:

$3 e^{4} - \frac{3}{e^{3}} - 28$

Decimal Form:

$135.64508889 \dots$

Step 8