Advertisements
Advertisements
Question
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
Options
`"a" - 1 + "e"/2`
`"a" + 1 - "e"/2`
`"a" - 1 - "e"/2`
`"a" + 1 + "e"/2`
Advertisements
Solution
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to `"a" + 1 - "e"/2`.
Explanation:
Since I = `int_0^1 "e"^"t"/(1 + "t") "dt"`
= `|1/(1 + "t") "e"^"t"|_0^1 + int_0^1 "e"^"t"/(1 + "t")^2 "dt"` = a ...(Given)
Therefore, `int_0^1 "e"^"t"/(1 + "t")^2 = "a" - "e"/2 + 1`.
APPEARS IN
RELATED QUESTIONS
If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`
(A) 1
(B) 2
(C) –1
(D) –2
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) cos^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`
Evaluate `int_0^1 x(1 - x)^5 "d"x`
`int_0^1 (1 - x/(1!) + x^2/(2!) - x^3/(3!) + ... "upto" ∞)` e2x dx = ?
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_(-pi/4)^(pi/4) 1/(1 - sinx) "d"x` = ______.
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
If `int_0^1(sqrt(2x) - sqrt(2x - x^2))dx = int_0^1(1 - sqrt(1 - y^2) - y^2/2)dy + int_1^2(2 - y^2/2)dy` + I then I equal.
`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.
The value of the integral `int_0^sqrt(2)([sqrt(2 - x^2)] + 2x)dx` (where [.] denotes greatest integer function) is ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
Evaluate `int_-1^1 |x^4 - x|dx`.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Solve the following.
`int_1^3 x^2 logx dx`
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate: `int_-1^1 x^17.cos^4x dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate:
`int_0^6 |x + 3|dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
`∫_0^(π/2) (sqrttan x + sqrtcot x)dx` = ______.
