What is the integral of cosh^4(7x) ?

The integral of cosh^4(7x) is 1/7 (1/4 cosh^3(7x)sinh(7x)+3/8 (7x+1/2 sinh(14x)))+C

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integral of cosh^4(7x)

\int cosh^{4} (7 x) d x

\frac{1}{7} (\frac{1}{4} cosh^{3} (7 x) sinh (7 x) + \frac{3}{8} (7 x + \frac{1}{2} sinh (14 x))) + C

Solution steps

\int cosh^{4} (7 x) d x

Apply u-substitution

: \int cosh^{4} (u) \frac{1}{7} d u

= \int cosh^{4} (u) \frac{1}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int cosh^{4} (u) d u

Apply Integral Reduction

= \frac{1}{7} (\frac{cosh ^{3} ( u ) sinh ( u )}{4} + \frac{3}{4} \cdot \int cosh^{2} (u) d u)

\int cosh^{2} (u) d u = \frac{1}{2} (u + \frac{1}{2} sinh (2 u))

= \frac{1}{7} (\frac{cosh ^{3} ( u ) sinh ( u )}{4} + \frac{3}{4} \cdot \frac{1}{2} (u + \frac{1}{2} sinh (2 u)))

Substitute back

u = 7 x

= \frac{1}{7} (\frac{cosh ^{3} ( 7 x ) sinh ( 7 x )}{4} + \frac{3}{4} \cdot \frac{1}{2} (7 x + \frac{1}{2} sinh (2 \cdot 7 x)))

Simplify

\frac{1}{7} (\frac{cosh ^{3} ( 7 x ) sinh ( 7 x )}{4} + \frac{3}{4} \cdot \frac{1}{2} (7 x + \frac{1}{2} sinh (2 \cdot 7 x))) : \frac{1}{7} (\frac{1}{4} cosh^{3} (7 x) sinh (7 x) + \frac{3}{8} (7 x + \frac{1}{2} sinh (14 x)))

= \frac{1}{7} (\frac{1}{4} cosh^{3} (7 x) sinh (7 x) + \frac{3}{8} (7 x + \frac{1}{2} sinh (14 x)))

Add a constant to the solution

= \frac{1}{7} (\frac{1}{4} cosh^{3} (7 x) sinh (7 x) + \frac{3}{8} (7 x + \frac{1}{2} sinh (14 x))) + C

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What is the integral of cosh^4(7x) ?
The integral of cosh^4(7x) is 1/7 (1/4 cosh^3(7x)sinh(7x)+3/8 (7x+1/2 sinh(14x)))+C