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If intf(x)sinxcosxdx=1/(2(b^2-a^2))lnf(x...

If `intf(x)sinxcosxdx=1/(2(b^2-a^2))lnf(x)+c` ,then f(x) is equal to

A

`(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)`

B

`(1)/(a^(2)sin^(2)x-b^(2)cos^(2)x)`

C

`(1)/(a^(2)sin^(2)x+b^(2)sin^(2)x)`

D

`(1)/(a^(2)cos^(2)x-b^(2)sin^(2)x)`

Text Solution

Verified by Experts

The correct Answer is:
a
Let `I=int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)sinxcosxdx`
`rArrI=(1)/(2(a^(2)-b^(2)))int((a^(2)-b^(2))sin2x)/(a^(2)sin^(2)x+b^(2)cos^(2)x)dx`
`rArr I=(1)/(2(a^(2)-b^(2)))int(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)d(a^(2)sin^(2)x+b^(2)cos^(2)x)`
`rArr I=(1)/(2(a^(2)-b^(2)))log|a^(2)sin^(2)x+b^(2)cos^(2)x|+C`
Hence , f (x) `=(1)/(a^(2)sin^(2)x+b^(2)cos^(2)x)`
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OBJECTIVE RD SHARMA ENGLISH-INDEFINITE INTEGRALS-Solved Example
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  2. Evaluate: int(x^7)/((1-x^2)^5)\ dx .

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  3. If intf(x)sinxcosxdx=1/(2(b^2-a^2))lnf(x)+c ,then f(x) is equal to

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  4. The value of int(dx)/(x^n(1+x^n)^(1/ n)) is equal to

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  8. int(e^((x^(2)+4Inx))-x^(3)e^(x^(2)))/(x-1)dx equals to

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  9. The value of the integral int(xsinx^(2)e^(secx^(2)))/(cos^(2)x^(2))dx ...

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  11. intsqrt(x-3)(sin^(-1)(Inx)+cos^(-1)(Inx))dx is equal to

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  12. int(1-x^(7))/(x(1+x^(7)))dx equals

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  13. Evaluate: (sin^3x dx)/((cos^4x+3cos^2x+1)tan^(-1)(secx+cosx)

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  14. int((x-x^5)^(1//5))/(x^6)dx is equal to :

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  15. int(tanx)/(sqrt(sin^(4)x+cos^(4)x))dx is equal to

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  16. If intsqrt((cos^(3)x)/(sin^(11)x))dx =-2(Atan^(-9/2)+Btan^(-5/2)x) + C...

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  17. int ((f (x) g' (x) -f' (x) g (x))/(f (x) g(x)))(log (g(x )) - log (f(x...

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  18. int ((f (x) g' (x) -f' (x) g (x))/(f (x) g(x)))(log (g(x )) - log (f(x...

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  19. int(x^(x))^(x)(2x log(e)x+x)dx is equal to

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  20. Let the equation of a curve passing through the point (0,1) be given b...

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