What is the integral of (tan^4(x)-sec^4(x)) ?

The integral of (tan^4(x)-sec^4(x)) is x(tan^4(x)-sec^4(x))+4(1/2 xtan^2(x)-1/2 (-x+tan(x)))+C

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integral of (tan^4(x)-sec^4(x))

\int (tan^{4} (x) - sec^{4} (x)) d x

x (tan^{4} (x) - sec^{4} (x)) + 4 (\frac{1}{2} x tan^{2} (x) - \frac{1}{2} (- x + tan (x))) + C

Solution steps

\int tan^{4} (x) - sec^{4} (x) d x

Apply Integration By Parts

: x (tan^{4} (x) - sec^{4} (x)) - \int - 4 x sec^{2} (x) tan (x) d x

= x (tan^{4} (x) - sec^{4} (x)) - \int - 4 x sec^{2} (x) tan (x) d x

\int - 4 x sec^{2} (x) tan (x) d x = - 4 (\frac{1}{2} x tan^{2} (x) - \frac{1}{2} (- x + tan (x)))

= x (tan^{4} (x) - sec^{4} (x)) - (- 4 (\frac{1}{2} x tan^{2} (x) - \frac{1}{2} (- x + tan (x))))

Simplify

= x (tan^{4} (x) - sec^{4} (x)) + 4 (\frac{1}{2} x tan^{2} (x) - \frac{1}{2} (- x + tan (x)))

Add a constant to the solution

= x (tan^{4} (x) - sec^{4} (x)) + 4 (\frac{1}{2} x tan^{2} (x) - \frac{1}{2} (- x + tan (x))) + C

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What is the integral of (tan^4(x)-sec^4(x)) ?
The integral of (tan^4(x)-sec^4(x)) is x(tan^4(x)-sec^4(x))+4(1/2 xtan^2(x)-1/2 (-x+tan(x)))+C