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If alpha=int(0)^(1) e^(3x+3tan^(-1)x((12...

If `alpha=int_(0)^(1) e^(3x+3tan^(-1)x((12+9x^(2))/(1+x^(2)))dx`, where `tan^(-1)` takes principal values, then the value of
`(log_(e )|1+alpha|-(3pi)/(4))`,is

A

9

B

8

C

7

D

6

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`alpha=underset(0)overset(1)int e^(3x+3tan^(-1)x)((12+9x^(2))/(1+x^(2)))dx`
`rArr alpha underset(0)overset(9+(3pi)/(4))int e^(t)dt`, where t=9x+3`tan^(-1)x`
`rArr alpha=e^(9+(3pi)/(4))-1`
`rArr log_(e )|1+alpha|=9+(3pi)/(4)rArr log_(e )|1+alpha|-(3pi)/(4)=9`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Chapter Test 2
  1. If alpha=int(0)^(1) e^(3x+3tan^(-1)x((12+9x^(2))/(1+x^(2)))dx, where t...

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  2. The integral int(0)^(r pi) sin^(2x)x dx is equal to

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  3. The value of the integral int(0)^(2)x[x]dx

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  4. The value of integral sum (k=1)^(n) int (0)^(1) f(k - 1+x) dx is

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  5. Let f(x) be a funntion satifying f'(x)=f(x) with f(0)=1 and g(x) be th...

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  6. If I=int(0)^(1) cos{ 2 "cot"^(-1)sqrt((1-x)/(1+x))}dx then

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  7. The value of int(a)^(a+(pi//2))(sin^(4)x+cos^(4)x)dx is

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  8. The vaue of int(-1)^(2) (|x|)/(x)dx is

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  9. The value of int0^1 (x^(3))/(1+x^(8))dx is

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  10. The value of int(0)^(3) xsqrt(1+x)dx, is

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  11. The value of the integral int(0)^(1) log sin ((pix)/(2))dx is

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  12. The value of the integral int(0)^(pi)x log sin x dx is

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  13. If I(1)=int(0)^(oo) (dx)/(1+x^(4))dx and I(2)underset(0)overset(oo)in...

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  14. If f(x)={{:(x,"for " x lt 1),(x-1,"for " x ge1):},"then" int(0)^(2) x...

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  15. The value of the integral int(0)^(2) (1)/((x^(2)+1)^(3//2))dx is

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  21. If int(pi//2)^(x) sqrt(3-2sin^(2)u) dx+int(dx)^(dy) equal pi//2

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