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If f(x) = tanx-tan ^(3) x + tan^(5) x - ...

If `f(x) = tanx-tan ^(3) x + tan^(5) x - tan ^(7) x + ... infty` for
`olt x lt pi/4 , "than" int_(0)^(pi//4) f (x) dx=`

A

1

B

0

C

`1/4`

D

`1/2`

Text Solution

Verified by Experts

The correct Answer is:
C
`f(x)=tanx-tan^(3)x+tan^(5)x-...infty` [ it is a Gp]
`rArrf(x)=(tanx)/(1+tan^(2)x)=(tanx)/(sec^(2)x)=(sin2x)/(2)`
`thereforeint_(0)^(pi//4)f(x)dx=int_(0)^(pi//4)(sin2x)/(2)dx=[-(cos2x)/(4)]_(0)^(pi//4)=(1)/(4)`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
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  2. Evaluate: ("lim")(nvecoo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))++1/(sqrt...

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  3. If f(x) = tanx-tan ^(3) x + tan^(5) x - tan ^(7) x + ... infty for o...

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  4. The value of int(0)^((pi)/(8))cos^(3)4 theta d theta is equal to -

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  5. int (0)^(pi//3) (cos x + sin x)/(sqrt(1+sin 2x))dx is equal to

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  6. int (0)^(1) x^(3//2) sqrt(1-x) dx is equal to

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  7. The value of int (1)^(2) (dx)/((x+1) sqrt((x^2)-1)) is

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  8. If f(x)=|(sin x+sin x 2x+sin3x,sin2x,sin3x),(3+4sinx,3,4sinx),(1+sinx,...

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  9. int (-1)^(1) (e^(x^(3)) +e^(-x^(3))) (e^(x)-e^(-x)) dx is equal to

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  10. underset(n to oo)lim((1)/(n)+(1)/(n+1)+...+(1)/(3n)) is equal to

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  11. Let f (x) = x – [x], for every real number x, where [x] is the greates...

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  12. The value of int (0)^(1) (x^(4) +1)/(x^(2)+1)dx is

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  13. The value of overset(5)underset(3)int (x^(2))/(x^(2)-4)dx, is

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  14. The value of int (0)^(pi//2) ((sin x + cos x)^(2))/(sqrt(1+sin 2x) dx...

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  15. int (0)^(a) sqrt(a^(2) - x^(2)) dx is equal to

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  16. The value of int(e^(-1))^(e) (dt)/(t(t+1)) is equal to

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  17. The value of int (1)^(e) 10^(log(e)x) dx is equal to

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  18. If int (2)^(e) (1/(logx)-1/(logx)^(2))dx = a + b/(log2) , then

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  19. int (0)^(oo) (dx)/((x+ sqrt(x^(2)+1))^(3)) is equal to

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  20. The value of the integral int (-pi//2)^(pi//2) sqrt(cos x - cos ^(3) ...

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