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int(1)/(tanx+cotx+secx+"cosec "x)dx is e...

`int(1)/(tanx+cotx+secx+"cosec "x)dx` is equal to

A

`(1)/(2)(sinx+cosx+x)+C`

B

`(1)/(2)(sinx-cosx-x)+C`

C

`(1)/(2)(cosx-xsinx)+C`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
d
Let `=int(1)/(tanx+cotx+secx+"cosec " x)dx` Then ,
`I=int(sinxcosx)/(1+sinx+cosx)dx`
`rArrI=(1)/(2)int((1+2sinx cosx-1))/(1+sinx+cosx)dx=(1)/(2)int((sinx+cosx)^(2)-1)/(1+sinx+cosx)`
`rArr I=(1)/(2)int((sinx+cosx+1)(sinx+cosx-1))/(1+sinx+cosx)dx`
`rArr I=(1)/(2)int(sinx+cosx-1)dx`
`rArrI=(1)/(2)(-cosx+sinx-1)+C`
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The given expression f(x)=(1)/(tanx+cotx+secx+"cosec x") is equivalent to

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int(secx."cosec"x)/(2cotx-secx"cosec x")dx di equal to

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OBJECTIVE RD SHARMA-INDEFINITE INTEGRALS-Solved Example
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  2. int e^(x^4) (x + x^3 +2x^5) e^(x^2) dx is equal to

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  3. int[sin(101 x)*sin^(99)x]dx

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  6. If int 2/(2-x)^2 ((2-x)/(2+x))^(1//3)\ dx = lambda ((2+x)/(2-x))^mu + ...

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  7. Let f(x) be a quadratic function such that f(0)=1 and int(f(x))/(x^2(x...

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

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  11. Let f (x) be a quadratic function such that f (0) =1 and f(-1)=4, if...

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  12. Evaluate: int(e^tan^((-1)x))/((1+x^2))[(sec^(-1)sqrt(1+x^2)+cos^(-1)((...

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  13. intx^(2)((xsec^(2)x+tanx))/((xtanx+1)^(2))dx=

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  14. int (mx^(m+2n-1)-nx^(n-1))/(x^(2m+2n)+2x^(m+n)+1)dx is equal to

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  15. int(1)/(tanx+cotx+secx+"cosec "x)dx is equal to

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  16. int(x^(2)(1-"In "x))/(("In "x)^(4)-x^(4))dx is equal to

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  17. The integral int(2x^(12)+5x^(9))/((x^(5)+x^(3)+1)^(3))dx is equal to ...

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  18. If int(1)/(cos^(3)xsqrt(2 sin2x))dx=(tanx)^(A)+C(tanx)^(B)+k, where k...

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  19. "If"int(dx)/(x^(3)(1+x^(6))^(2/3))=f(x)(1+x^(6))^(1/3)+C where, C is a...

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  20. The integral int(1)/((1+sqrt(x))sqrt(x-x^(2)))dx is equal to (where C ...

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