Advertisements
Advertisements
प्रश्न
\[\int\limits_0^1 \log\left( 1 + x \right) dx\]
Advertisements
उत्तर
\[\int_0^1 \log\left( 1 + x \right) d x\]
\[ = \int_0^1 \log\left( 1 + x \right) \times 1 d x\]
\[ = \left[ \log\left( 1 + x \right) x \right]_0^1 - \int_0^1 \frac{x}{1 + x}dx\]
\[ = \left[ \log\left( 1 + x \right) x \right]_0^1 - \int_0^1 \left( 1 - \frac{1}{1 + x} \right)dx\]
\[ = \left[ x\log\left( 1 + x \right) \right]_0^1 - \left[ x - \log\left( 1 + x \right) \right]_0^1 \]
\[ = \log2 - 1 + \log2\]
\[ = 2\log2 - 1\]
\[ = \log4 - \log e\]
\[ = \log\frac{4}{e}\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]
If f(x) is a continuous function defined on [−a, a], then prove that
Solve each of the following integral:
If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]
The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is
The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is
\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]
\[\int\limits_0^1 x \left( \tan^{- 1} x \right)^2 dx\]
\[\int\limits_0^\pi x \sin x \cos^4 x dx\]
\[\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx\]
\[\int\limits_0^{\pi/2} \frac{dx}{4 \cos x + 2 \sin x}dx\]
Using second fundamental theorem, evaluate the following:
`int_0^(pi/2) sqrt(1 + cos x) "d"x`
Evaluate the following:
`int_1^4` f(x) dx where f(x) = `{{:(4x + 3",", 1 ≤ x ≤ 2),(3x + 5",", 2 < x ≤ 4):}`
Evaluate the following using properties of definite integral:
`int_(-1)^1 log ((2 - x)/(2 + x)) "d"x`
Evaluate the following:
`int_0^oo "e"^(-4x) x^4 "d"x`
Evaluate the following integrals as the limit of the sum:
`int_0^1 (x + 4) "d"x`
If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4ex + 5e –x| + C, then ______.
If `intx^3/sqrt(1 + x^2) "d"x = "a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"`, then ______.
Find: `int logx/(1 + log x)^2 dx`
Evaluate \[\int_{0}^{1}x\,dx\].
