CBSE Class 12-science Answered
maths que tell
![question image](http://images.topperlearning.com/topper/new-ate/top_mob1699270406876564916c34dd709-91bf-4698-8b5e-b27b2d1e7585.jpg)
Asked by manyarajawat1 | 06 Nov, 2023, 17:03: PM
It is expected that student should post only one question in one posting. But the image shows huge number of questions.
Hence only first question is solved .
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![begin mathsize 14px style I space equals space integral subscript 0 superscript pi divided by 2 end superscript square root of 1 minus cos left parenthesis 2 x right parenthesis end root space d x end style](https://images.topperlearning.com/topper/tinymce/cache/2bdd9f41624f22be84be64b36fe91dd0.png)
1- cos(2x) = 2 sin2(x)
Hence , above integration becomes
![begin mathsize 14px style I space equals space square root of 2 space integral subscript 0 superscript pi divided by 2 end superscript space sin left parenthesis x right parenthesis space d x space space equals space minus square root of 2 space open parentheses cos space x close parentheses subscript 0 superscript straight pi divided by 2 end superscript space equals space square root of 2 end style](https://images.topperlearning.com/topper/tinymce/cache/cda281669223c6ab9aee160062802b6e.png)
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![begin mathsize 14px style I space equals space integral x space tan squared x space d x end style](https://images.topperlearning.com/topper/tinymce/cache/75b056b5a8da8c7fda573c81f0a0b0be.png)
Le t u = x and dv = tan2x dx
Hence du = dx and v = ![begin mathsize 14px style integral tan squared x space d x end style](https://images.topperlearning.com/topper/tinymce/cache/a374e717e461fd2e6c7984d3a2063058.png)
![begin mathsize 14px style integral tan squared x space d x end style](https://images.topperlearning.com/topper/tinymce/cache/a374e717e461fd2e6c7984d3a2063058.png)
![begin mathsize 14px style integral tan squared x space d x space equals space integral space left parenthesis space s e c squared x space minus 1 space right parenthesis space d x space equals space integral s e c squared x space d x space minus space integral d x space equals space tan left parenthesis x right parenthesis space minus space x end style](https://images.topperlearning.com/topper/tinymce/cache/e18ab8b2e5bb382121b42730fe03fb73.png)
Hence v = tan(x) - x
Hence integration given in eqn.(1) is solved as
![begin mathsize 14px style I space equals space x space left parenthesis space tan space x space minus space x space right parenthesis space minus integral left parenthesis tan space x space minus space x space right parenthesis space d x end style](https://images.topperlearning.com/topper/tinymce/cache/ce9d90d61fe763d7be51d50a24bbc500.png)
![begin mathsize 14px style I space equals space x space left parenthesis space tan space x space minus space x space right parenthesis space plus space log left parenthesis cos x right parenthesis space plus space x squared over 2 end style](https://images.topperlearning.com/topper/tinymce/cache/253e2f432d3e82042135b052a08d09af.png)
Answered by Thiyagarajan K | 06 Nov, 2023, 23:09: PM
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