What is the integral of y^2cosh(7y) ?

The integral of y^2cosh(7y) is 1/343 (49y^2sinh(7y)-2(7ycosh(7y)-sinh(7y)))+C

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integral of y^2cosh(7y)

\int y^{2} cosh (7 y) d y

\frac{1}{343} (49 y^{2} sinh (7 y) - 2 (7 y cosh (7 y) - sinh (7 y))) + C

Solution steps

\int y^{2} cosh (7 y) d y

Apply u-substitution

: \int \frac{u ^{2} cosh ( u )}{343} d u

= \int \frac{u ^{2} cosh ( u )}{343} d u

Take the constant out:

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

= \frac{1}{343} \cdot \int u^{2} cosh (u) d u

Apply Integration By Parts

: u^{2} sinh (u) - \int 2 u sinh (u) d u

= \frac{1}{343} (u^{2} sinh (u) - \int 2 u sinh (u) d u)

\int 2 u sinh (u) d u = 2 (u cosh (u) - sinh (u))

= \frac{1}{343} (u^{2} sinh (u) - 2 (u cosh (u) - sinh (u)))

Substitute back

u = 7 y

= \frac{1}{343} ((7 y)^{2} sinh (7 y) - 2 (7 y cosh (7 y) - sinh (7 y)))

Simplify

\frac{1}{343} ((7 y)^{2} sinh (7 y) - 2 (7 y cosh (7 y) - sinh (7 y))) : \frac{1}{343} (49 y^{2} sinh (7 y) - 2 (7 y cosh (7 y) - sinh (7 y)))

= \frac{1}{343} (49 y^{2} sinh (7 y) - 2 (7 y cosh (7 y) - sinh (7 y)))

Add a constant to the solution

= \frac{1}{343} (49 y^{2} sinh (7 y) - 2 (7 y cosh (7 y) - sinh (7 y))) + C

x \to 0 lim (\frac{x}{4 ^{x} - 2 ^{x}})

\int_{0}^{7} \frac{8 t}{( t - 8 ) ^{2}} d t

x \to 0 - lim (x \cdot sin (\frac{1}{x}))

\frac{d}{d t} (- e^{- t})

\frac{d}{d x} (x^{2} 7^{x})

What is the integral of y^2cosh(7y) ?
The integral of y^2cosh(7y) is 1/343 (49y^2sinh(7y)-2(7ycosh(7y)-sinh(7y)))+C