Advertisements
Advertisements
Question
Prove the following:
`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`
Advertisements
Solution
`int_0^(pi/4) 2 tan^3 x dx`
`= int_0^(pi/4) 2 tan x* tan^2 x dx`
`= int_0^(pi/4) 2 tan x (sec^2 x - 1) dx`
`= 2 int_0^(pi/4) (tan x) sec^2 x dx - 2 int_0^(pi/2) tan x dx`
`= 2 [(tan^2 x)/2]_0^(pi/4) - 2 [- log |cos x|] _0^(pi/4)`
`= (tan^2 pi/4 - tan^2 0) + 2 (log cos pi/4 - log cos 0)`
`= (1 - 0) + 2 (log 1/ sqrt2 - log 1)`
`= 1 + 2 (log 1 - log sqrt 2 - log 1)`
`= 1 + 2 xx (-1/2 log 2)`
= 1 - log 2
APPEARS IN
RELATED QUESTIONS
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums `int_(-1)^1 e^x dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_(pi/6)^(pi/3) (sin x + cosx)/sqrt(sin 2x) dx`
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Evaluate the definite integral:
`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`
Prove the following:
`int_1^3 dx/(x^2(x +1)) = 2/3 + log 2/3`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
Choose the correct answers The value of `int_0^1 tan^(-1) (2x -1)/(1+x - x^2)` dx is
(A) 1
(B) 0
(C) –1
(D) `pi/4`
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate the following integrals as limit of sums:
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0
Evaluate:
`int (sin"x"+cos"x")/(sqrt(9+16sin2"x")) "dx"`
If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to ______.
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
The value of `int_(-pi)^pi sin^3x cos^2x "d"x` is ______.
Left `f(x) = {{:(1",", "if x is rational number"),(0",", "if x is irrational number"):}`. The value `fof (sqrt(3))` is
The limit of the function defined by `f(x) = {{:(|x|/x",", if x ≠ 0),(0",", "otherwisw"):}`
`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4` is equal to
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
