Find the value of `int_(-1)^1[x^2+{x}]dx ,w h e r e[dot]a n d{dot}` denote the greatest function and fractional parts of `x ,` respectively.
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Evaluate: `int_(-5)^5x^2[x+1/2]dx(w h e r e[dot]` denotes the greatest integer function).
Let `I=int_(-5)^(5)x^(2)[x+1/2]dx` `=int_(-5)^(5)x^(2)(-x+1/2]dx` (Replacing `x` by `-x`) `=int_(-5)^(5)x^(2)[-x-1/2+1]dx` `=int_(-5)^(5)([-x-1/2]+1)dx` `=-int_(5)^(5)x^(2)[x+1/2]dx` `=-I` `:.I=0`
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`I_(1)=int_(0)^((pi)/2)(sinx-cosx)/(1+sinxcosx)dx, I_(2)=int_(0)^(2pi)cos^(6)dx`, `I_(3)=int_(-(pi)/2)^((pi)/2)sin^(3)xdx, I_(4)=int_(0)^(1) In (1/x-1
Correct Answer - C `I_(1)=int_(0)^(pi//2)(sinx_cosx)/(1+sinx cos) dx` `=int_(0)^(pi//2) ("sin"((pi)/2-x)-"cos"((pi)/2-x))/(1+"sin"((pi)/2-x)"cos"((pi)/2-x))` `=int_(0)^(pi//2)(cosx-sinx)/(1+sinx cosx)dx=-I_(1)` or `I_(1)=0` `I_(3)=0` as `sin^(3)x` is odd `I_(4)=int_(0)^(1)In((1-x)/x)dx` `=int_(0)^(1)In((1-(1-x))/(1-x))dx` `=int_(0)^(1)In x/(1-x)dx=-I_(4)` or `I_(4)=0` `I_(2)=int_(0)^(2pi)cos^(6)x dx=2int_(0)^(pi)cos^(6)x dx!=0`
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If `int_(0)^(x)[x]dx=int_(0)^(|x|)x dx,` (where [.] and {.} denotes the greatest integer and fractional part respectively), then
Correct Answer - A::B `int_(0)^(x)[x]dx=int_(0)^(1)0dx+int_(1)^(2)1dx+int_(2)^(3)2dx+…………….` `+int_([x]-1)^([x])([x]-1)dx+int_([x])^(x)[x]dx` `=0+1+2+3+……….+([x]-1)+[x](x-[x])` `=(([x]-1)[x])/2+[x]{x}`…………..1 Now `int_(0)^([x]) dx=[(x^(2))/2]_(0)^([x])=([x]^(2))/2`..............2 Comparing 1 and 2 we get `([x]-1)[x])/2+[x]{x}=([x]^(2))/2` `:.[x]^(2)-[x]+2[x]{x}=[x]^(2)` `=[x]=0` or `{x}=1//2`
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Evaluate : (i) `int_(-oo)^(oo) (dx)/(x^(2)+2x+2)` , (ii) `int_(sqrt(2))^(oo)(dx)/(xsqrt(x^(2)-1))`, (iii) `int_(0)^(4)(x^(2))/(1+x)dx` (iv) `int_(0)^(
Correct Answer - (i) `pi` , (ii) `pi/4` , (iii) `4 + ln 5` , (iv) `8/21`
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Evaluate : (i) Find the value a such `int_(0)^(a)(1)/(e^(x)+4e^(x)+5)dx = ln 3sqrt(2)`. (ii) Find the value of `int_(0)^((pi//2)^(1//2))x^(5).sinx^(3)
Correct Answer - (i) `ln 11` , (ii) `1/3`
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Evaluate : (i) `int_(0)^(4)[x]^(2) dx` (where `[*]` denotes greatest integer function) (ii) `int_(0)^(pi)sqrt(1+sin2x)dx` , (iii) `int_(0)^(2)f(x)dx`
Correct Answer - (i) `5-sqrt(2) - sqrt(3)` , (ii) `2sqrt(2)`, (iii) `9` , (iv) `4` , (v) `cot 1` , (vi) `29` (vii) `cos 1+cos2 + cos 3 + 0`
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Evaluate : (i) `int_(-1)^(1)e^(x)dx` , (ii) `int_(-pi//4)^(pi//4)|sinx|dx` , (iii) `int_(-pi//4)^(pi//4)(x+pi//4)/(2-cos2x)dx` (iv) `int_(-1)^(1)sin^(
Correct Answer - (i) `2e-2` , (ii) `2-sqrt(2)`, (iii) `(pi^(2))/(6sqrt(3))` , (iv) `0` , ( v) `0`
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Evaluate : (i) `int_(0)^(2pi) {sin(sinx)+sin(cosx)}dx`, (ii) `int_(0)^(pi) (dx)/(5+4cos2x)` (iii) `int_(0)^(pi//2) (2lnsinx-ln sin2x)dx` , (iv) `int_(
Correct Answer - (i) `0` , (ii) `pi/3` , (iii) `- (pi)/(2) ln 2` , (iv) `piln 2`
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Evaluate : (i) `int_(-1)^(2){2x}dx` (where function`{*}` denotes fractional part function) (ii) `int_(0)^(10x)(|sinx|+|cosx|) dx` (iii) `(int_(0)^(n)[
Correct Answer - (i) `3/2` , (ii) `40` , (iii) `n-1` , (iv) `4n`
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Evaluate : (i) `int_(-pi//2)^(pi//2)sin^(2)xcos^(2)x(sinx+cosx)dx` , (ii) `int_(0)^(pi)xsin^(5)xdx` (iii) `int_(0)^(2)x^(3//2)sqrt(2-x)dx`, (iv) `int_
Correct Answer - (i) `4/15` , (ii) `pi/2`, (iv) `(pi^(2))/(4)`
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