CBSE Class 12-science Answered
Solve it pls.
![question image](http://images.topperlearning.com/topper/new-ate/top_mob16960946671623488695b85c63f2-1d6b-4d03-9fac-6022b6da7b5d.jpg)
Asked by sdmbotch1123 | 30 Sep, 2023, 22:54: PM
Integration can be solved only if
![begin mathsize 14px style I space equals space integral fraction numerator square root of tan space x end root space over denominator cos to the power of 4 x end fraction space d x end style](https://images.topperlearning.com/topper/tinymce/cache/c328b8d697bed77ef3a0e9f9ccd7eaaa.png)
Above integration is solved by rewriting above expression as follows
![begin mathsize 14px style I space equals space integral square root of tan space x end root space s e c squared x space space s e c squared x space d x end style](https://images.topperlearning.com/topper/tinymce/cache/050e7ae4c9b0757373b36cf921f50b4a.png)
let us use the substitution u = tan(x)
then, we have, du = sec2 (x) dx
sec2 (x) = 1 + tan2 (x) = 1 + u2
hence expression (1) becomes
![begin mathsize 14px style I space equals space integral space square root of u space left parenthesis space 1 space plus space u squared space right parenthesis space d u space space equals space integral space u to the power of 3 divided by 2 end exponent space d u space plus space integral space u to the power of 5 divided by 2 end exponent space d u end style](https://images.topperlearning.com/topper/tinymce/cache/a503cd414339fef141f4bdbeb50340fa.png)
![begin mathsize 14px style I space equals space 2 over 5 u to the power of 5 divided by 2 end exponent space plus space 2 over 7 space u to the power of 7 divided by 2 end exponent space space equals space 2 space u to the power of 5 divided by 2 end exponent space open parentheses 1 fifth space plus space 1 over 7 space u close parentheses end style](https://images.topperlearning.com/topper/tinymce/cache/c515c8e71172d629cea407ddc2344f76.png)
By resubstituting u = tanx , we get
![begin mathsize 14px style I space equals space 2 space left parenthesis square root of tan space x end root space right parenthesis to the power of 5 space open parentheses 1 fifth space plus space 1 over 7 tan x close parentheses end style](https://images.topperlearning.com/topper/tinymce/cache/e871f0731c7787522fb668b476ccdcc6.png)
Answered by Thiyagarajan K | 01 Oct, 2023, 08:58: AM
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