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Grade 12Integral Calculus

Integrate log(sinx+cosx)dx from -pi/4 to pi/4. someone plz help

Saboor

10 Years agoGrade 12

2 Answers

Riddhish Bhalodia

10 Years ago

This is a tricky question
First put x = -x and add
$I = \int \limits_{-\pi/4}^{\pi/4}log(sinx+cosx)dx = \int \limits_{-\pi/4}^{\pi/4}log(-sinx+cosx)dx$
$2I = \int \limits_{-\pi/4}^{\pi/4}log(-sin^2x+cos^2x)dx = \int \limits_{-\pi/4}^{\pi/4}log(cos2x)dx = 2\int \limits_{0}^{\pi/4}log(cos2x)dx$ $I =\int \limits_{0}^{\pi/4}log(cos2x)dx$
Now put 2x = t which implies, dx = dt/2
$I =1/2\int \limits_{0}^{\pi/2}log(cost)dt$
Now refer to the following steps to solve for the simplified integral

xfg

8 Years ago

what is first step?

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