Advertisements
Advertisements
Question
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
Options
log 2
2 log 2
`1/2 log 2`
4 log 2
Advertisements
Solution
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to 2 log 2.
Explanation:
Since I = `int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x`
= `int_(-1)^1 x^3/(x^2 + 2|x| + 1) + int_(-1)^1 (|x| + 1)/(x^2 + 2|x| + 1)"d"x`
= `0 + 2 int_0^1 (|x| + 1)/((|x| + 1)^2) "d"x` ....[odd function + even function]
= `2 int_0^1 (x + 1)/(x + 1)^2 "d"x`
= `2 int_0^1 1/(x + 1) "d"x`
= `2|log|x + 1|]_0^1`
= 2 log 2.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
The total revenue R = 720 - 3x2 where x is number of items sold. Find x for which total revenue R is increasing.
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^{pi/2} log(tanx)dx` = ______
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
`int_(-2)^2 |x cos pix| "d"x` is equal to ______.
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
`int (dx)/(e^x + e^(-x))` is equal to ______.
The value of `int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2)) dx` is
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
If f(x) = `(2 - xcosx)/(2 + xcosx)` and g(x) = logex, (x > 0) then the value of the integral `int_((-π)/4)^(π/4) "g"("f"(x))"d"x` is ______.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
Select the formula for \[P_2\] : Splitting the Interval.
Which formula is \[P_5\] : Halving the Upper Limit (Addition)?
For an Even Function, which condition is satisfied?
If \[I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{\tan x}}\], what is the value of \[I\]?
