What is the integral from-ln(2) to 0 of cosh^2(x/2) ?

The integral from-ln(2) to 0 of cosh^2(x/2) is 1/2 (ln(2)+3/4)

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integral from-ln(2) to 0 of cosh^2(x/2)

\int_{- l n (2)}^{0} cosh^{2} (\frac{x}{2}) d x

\frac{1}{2} (ln (2) + \frac{3}{4})

Decimal

0.72157 \dots

Solution steps

\int_{- l n (2)}^{0} cosh^{2} (\frac{x}{2}) d x

Rewrite using trig identities

= \int_{- l n (2)}^{0} \frac{1}{2} (1 + cosh (x)) 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_{- l n (2)}^{0} 1 + cosh (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_{- l n (2)}^{0} 1 d x + \int_{- l n (2)}^{0} cosh (x) d x)

\int_{- l n (2)}^{0} 1 d x = ln (2)

\int_{- l n (2)}^{0} cosh (x) d x = \frac{3}{4}

= \frac{1}{2} (ln (2) + \frac{3}{4})

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\int_{- 4}^{4} 4 + 16 - x^{2} d x

What is the integral from-ln(2) to 0 of cosh^2(x/2) ?
The integral from-ln(2) to 0 of cosh^2(x/2) is 1/2 (ln(2)+3/4)