Denebel
Use the given trig identity to set up a usubstitution and then evaluate the indefinite integral.



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Denebel
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dw:1326790438109:dw

FoolForMath
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Yes, where are you stuck?

imperialist
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Hint: what is the derivative of tan(x)?

Denebel
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How to start? Can I do this ? dw:1326790556349:dw

Denebel
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Derivative of tan x is (sec x)^2

imperialist
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Yes, so try only breaking one of the (sec x)^2 into 1+(tan x)^2

imperialist
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You will be pleasantly surprised!

FoolForMath
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\[ \int \sec^4 x \space dx = \int (1+\tan^2 x) \sec^2 x \space dx \]

FoolForMath
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Now put \( \tan x = z \implies \sec^2 x dx = dz \)

FoolForMath
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So, \[ \int (1+\tan^2 x) \sec^2 x \space dx = \int (1+z^2) dz \] I am sure you can proceed after this.

Denebel
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Oh I see now. Thank you very much!

FoolForMath
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Glad to help :)

imperialist
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Likewise :)