Question

integrate..

step by step ✔✔

Answer 1

both arctanx and 1+x2, since 1+tan2t simplifies to sec2t=1cos2t,  as does tan′t.

Then try an integration by parts with regard to t (on one hand) and the trigonometric

part (on the other hand). This implies evaluating the anti-derivative oftan4tcost = sin4tcos5t = sin4t⋅costcos6t = sin4t⋅sin′t(1−sin2t)3,which is trivial, following a simple trigonometric substitution and some partial fraction

decomposition. The non-trivial part consists in integrating ln(1±sint),which, for all I

care, can be done by expanding the integrand into its Mercator series, and reversing the

order of summation and integration. Alternately, transform sint into cos2u using the

fact that sint=cos(π2−t), and then employ the well-known trigonometric formulas

for 1±cos2u, along with the properties of the natural logarithm. Now use the fact that

∫π40lncotu du =  Catalan's constant, and we are left with expressing∫π80lncotu du = ∫π80lncosu du−∫

Answer 2

Step-by-step explanation:

here is your answer

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