What is the integral of xcosh(2x^2-3) ?

The integral of xcosh(2x^2-3) is 1/4 sinh(2x^2-3)+C

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integral of xcosh(2x^2-3)

\int x cosh (2 x^{2} - 3) d x

\frac{1}{4} sinh (2 x^{2} - 3) + C

Solution steps

\int x cosh (2 x^{2} - 3) d x

Apply u-substitution

: \int \frac{cosh ( 2 u - 3 )}{2} d u

= \int \frac{cosh ( 2 u - 3 )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int cosh (2 u - 3) d u

Apply u-substitution

: \int cosh (v) \frac{1}{2} d v

= \frac{1}{2} \cdot \int cosh (v) \frac{1}{2} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int cosh (v) d v

Use the common integral:

\int cosh (v) d v = sinh (v)

= \frac{1}{2} \cdot \frac{1}{2} sinh (v)

Substitute back

= \frac{1}{2} \cdot \frac{1}{2} sinh (2 x^{2} - 3)

Simplify

\frac{1}{2} \cdot \frac{1}{2} sinh (2 x^{2} - 3) : \frac{1}{4} sinh (2 x^{2} - 3)

= \frac{1}{4} sinh (2 x^{2} - 3)

Add a constant to the solution

= \frac{1}{4} sinh (2 x^{2} - 3) + C

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

\int_{0}^{π} 2 sin^{4} (x) d x

\int \frac{x ^{3}}{x ^{2} + x + 1} d x

x^{2} y^{^{''}} - 7 x y^{^{'}} - 20 y = 0

y^{^{''}} + 9 y = 30 sin (t)

What is the integral of xcosh(2x^2-3) ?
The integral of xcosh(2x^2-3) is 1/4 sinh(2x^2-3)+C