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We calculate the definite integral of ln(cos x) over the interval from 0 to pi/2.Playlist: https://youtube.com/playlist?list=PL22w63XsKjqzJpcuD6InKWZXep2L0z1.

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1 Answer Jim H Aug 30, 2015 Integrate by parts (twice). Explanation: Looking at ∫cos(lnx)dx, we realize that if can't integrate straight away. Next thought is, perhaps we could use substitution. We would need cos(lnx) 1 x to integrate by substitution.

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Solved d/dx cos(x1/2) d/dxecos(x2) d/dx cos(ln(x + 1))

so basically the derivative of a function has the same domain as the function itself. Therefore the derivative of the function f (x)= ln (x), which is defined only of x > 0, is also defined only for x > 0 (f' (x) = 1/x where x > 0). i hope this makes sense. ( 2 votes)

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Figure 7.1.1: To find the area of the shaded region, we have to use integration by parts. For this integral, let's choose u = tan − 1x and dv = dx, thereby making du = 1 x2 + 1 dx and v = x. After applying the integration-by-parts formula (Equation 7.1.2) we obtain. Area = xtan − 1x|1 0 − ∫1 0 x x2 + 1 dx.

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Figure 6.2.7: Setting up Integration by Parts. Putting this all together in the Integration by Parts formula, things work out very nicely: $$\int \ln x\,dx = x\ln x - \int x\,\frac1x\,dx.\] The new integral simplifies to ∫ 1dx, which is about as simple as things get. Its integral is x + C and our answer is.

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