CBSE Class 12-science Answered

maths que tell

Asked by manyarajawat1 | 06 Nov, 2023, 05:03: PM

Expert Answer

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

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

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begin mathsize 14px style I space equals space integral x space tan squared x space d x end style

............................(1)

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

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

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

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

Answered by Thiyagarajan K | 06 Nov, 2023, 11:09: PM

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