What is the integral from-1 to 1 of cosh^2(x) ?

The integral from-1 to 1 of cosh^2(x) is (4e^2+e^4-1)/(4e^2)

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integral from-1 to 1 of cosh^2(x)

\int_{- 1}^{1} cosh^{2} (x) d x

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

Decimal

2.81343 \dots

Solution steps

\int_{- 1}^{1} cosh^{2} (x) d x

Rewrite using trig identities

= \int_{- 1}^{1} \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

= \frac{1}{2} \cdot \int_{- 1}^{1} 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

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

\int_{- 1}^{1} 1 d x = 2

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

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

Simplify

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

= \frac{4 e ^{2} + e ^{4} - 1}{4 e ^{2}}

What is the integral from-1 to 1 of cosh^2(x) ?
The integral from-1 to 1 of cosh^2(x) is (4e^2+e^4-1)/(4e^2)