Advertisements
Advertisements
प्रश्न
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
उत्तर
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
\[Let \log x = t\]
\[ \Rightarrow \frac{1}{x} dx = dt\]
\[Now, \int\frac{1}{x} \left( \log x \right)^2 dx\]
\[ = \int t^2 dt\]
\[ = \frac{t^3}{3} + C\]
\[ = \frac{\left( \log x \right)^3}{3} + C\]
APPEARS IN
संबंधित प्रश्न
Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums
Evaluate the following definite integrals as limit of sums.
`int_a^b x dx`
Evaluate the following definite integrals as limit of sums.
`int_2^3 x^2 dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
Evaluate the definite integral:
`int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
Prove the following:
`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`
Prove the following:
`int_0^1sin^(-1) xdx = pi/2 - 1`
`int dx/(e^x + e^(-x))` 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`
Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.
Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0
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 as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Evaluate the following:
`int_0^2 ("d"x)/("e"^x + "e"^-x)`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
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 ______.
If f" = C, C ≠ 0, where C is a constant, then the value of `lim_(x -> 0) (f(x) - 2f (2x) + 3f (3x))/x^2` is
Left `f(x) = {{:(1",", "if x is rational number"),(0",", "if x is irrational number"):}`. The value `fof (sqrt(3))` is
What is the derivative of `f(x) = |x|` at `x` = 0?
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
