What is the integral from-3 to 3 of (e^x-e^{-x})^2 ?

The integral from-3 to 3 of (e^x-e^{-x})^2 is (e^{12}-1)/(e^6)-12

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integral from-3 to 3 of (e^x-e^{-x})^2

\int_{- 3}^{3} (e^{x} - e^{- x})^{2} d x

\frac{e ^{12} - 1}{e ^{6}} - 12

Decimal

391.42631 \dots

Solution steps

\int_{- 3}^{3} (e^{x} - e^{- x})^{2} d x

Rewrite using trig identities

= \int_{- 3}^{3} (2 sinh (x))^{2} d x

Simplify

(2 sinh (x))^{2} : 4 sinh^{2} (x)

= \int_{- 3}^{3} 4 sinh^{2} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int_{- 3}^{3} sinh^{2} (x) d x

Rewrite using trig identities

= 4 \cdot \int_{- 3}^{3} \frac{- 1 + cosh ( 2 x )}{2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \frac{1}{2} \cdot \int_{- 3}^{3} - 1 + cosh (2 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 \cdot \frac{1}{2} (- \int_{- 3}^{3} 1 d x + \int_{- 3}^{3} cosh (2 x) d x)

\int_{- 3}^{3} 1 d x = 6

\int_{- 3}^{3} cosh (2 x) d x = \frac{e ^{12} - 1}{2 e ^{6}}

= 4 \cdot \frac{1}{2} (- 6 + \frac{e ^{12} - 1}{2 e ^{6}})

Simplify

4 \cdot \frac{1}{2} (- 6 + \frac{e ^{12} - 1}{2 e ^{6}}) : \frac{e ^{12} - 1}{e ^{6}} - 12

= \frac{e ^{12} - 1}{e ^{6}} - 12

What is the integral from-3 to 3 of (e^x-e^{-x})^2 ?
The integral from-3 to 3 of (e^x-e^{-x})^2 is (e^{12}-1)/(e^6)-12