Advertisements
Advertisements
Question
Advertisements
Solution
\[\int\frac{dx}{\sqrt{\tan^{- 1} x} \left( 1 + x^2 \right)}\]
\[Let \tan^{- 1} x = t\]
\[ \Rightarrow \frac{1}{1 + x^2} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{1}{1 + x^2}dx = dt\]
\[Now, \int\frac{dx}{\sqrt{\tan^{- 1} x} \left( 1 + x^2 \right)}\]
\[ = \int \frac{dt}{\sqrt{t}}\]
\[ = \int t^{- \frac{1}{2}} dt\]
\[ = \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + C\]
\[ = 2 \sqrt{t} + C\]
\[ = 2 \sqrt{\tan^{- 1} x} + C\]
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_a^b x dx`
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) dx`
Evaluate the following definite integrals as limit of sums.
`int_1^4 (x^2 - x) dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Prove the following:
`int_0^1 xe^x dx = 1`
Prove the following:
`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.
(A) `1/2`
(B) `1/3`
(C) `1/4`
(D) `1/5`
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate the following integral:
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`
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
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
Let f: (0,2)→R be defined as f(x) = `log_2(1 + tan((πx)/4))`. Then, `lim_(n→∞) 2/n(f(1/n) + f(2/n) + ... + f(1))` is equal to ______.
`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.
`lim_(n rightarrow ∞)1/2^n [1/sqrt(1 - 1/2^n) + 1/sqrt(1 - 2/2^n) + 1/sqrt(1 - 3/2^n) + ...... + 1/sqrt(1 - (2^n - 1)/2^n)]` is equal to ______.
