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What is the integral of Sec^3 4x

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The integral of sec^3(4x) can be found using the substitution method.

Let u = 4x, then du/dx = 4 and dx = du/4

Substitute these values into the integral:

  • ∫ sec^3(4x) dx = ∫ sec^3(u) (du/4)

Now we can use the formula for the integral of sec^3(x):

  • ∫ sec^3(x) dx = (1/2) sec(x) tan(x) + (1/2) ln |sec(x) + tan(x)| + C

Substitute back in u for x:

  • ∫ sec^3(4x) dx = (1/8) sec(4x) tan(4x) + (1/8) ln |sec(4x) + tan(4x)| + C

Therefore, the integral of sec^3(4x) is (1/8) sec(4x) tan(4x) + (1/8) ln |sec(4x) + tan(4x)| + C, where C is the constant of integration.

~ Zeph

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