Advertisements
Advertisements
प्रश्न
\[\int\limits_0^1 \cot^{- 1} \left( 1 - x + x^2 \right) dx\]
Advertisements
उत्तर
\[\int_0^1 co t^{- 1} \left( 1 - x + x^2 \right) d x\]
\[ = \int_0^1 co t^{- 1} \left[ x\left( x - 1 \right) + 1 \right] d x\]
\[ = \int_0^1 co t^{- 1} \left[ \frac{\left( x\left( x - 1 \right) + 1 \right)}{x - \left( x - 1 \right)} \right] d x\]
\[ = \int_0^1 co t^{- 1} x - co t^{- 1} \left( x - 1 \right) dx\]
\[ = \left[ xco t^{- 1} x \right]_0^1 + \int_0^1 \frac{x}{1 + x^2}dx - \left[ \left( x - 1 \right)co t^{- 1} \left( x - 1 \right) \right]_0^1 - \int_0^1 \frac{\left( x - 1 \right)}{1 + \left( x - 1 \right)^2}dx\]
\[ = \left[ xco t^{- 1} x \right]_0^1 + \frac{1}{2} \left[ \log\left( 1 + x^2 \right) \right]_0^1 - \left[ \left( x - 1 \right)co t^{- 1} \left( x - 1 \right) \right]_0^1 - \frac{1}{2} \left[ \log\left( 1 + \left( 1 - x \right)^2 \right) \right]_0^1 \]
\[ = \frac{\pi}{4} - \frac{1}{2}\log2 + \frac{\pi}{4} - \frac{1}{2}\log2\]
\[ = \frac{\pi}{2} - \log2\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following integral:
If f(x) is a continuous function defined on [−a, a], then prove that
If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to
Evaluate : \[\int\limits_0^\pi/4 \frac{\sin x + \cos x}{16 + 9 \sin 2x}dx\] .
`int_0^(2a)f(x)dx`
\[\int\limits_0^4 x\sqrt{4 - x} dx\]
\[\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]
\[\int\limits_0^1 \sqrt{\frac{1 - x}{1 + x}} dx\]
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
\[\int\limits_0^{2\pi} \cos^7 x dx\]
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
\[\int\limits_0^\pi \cos 2x \log \sin x dx\]
\[\int\limits_0^3 \left( x^2 + 1 \right) dx\]
Using second fundamental theorem, evaluate the following:
`int_0^3 ("e"^x "d"x)/(1 + "e"^x)`
Evaluate the following:
`int_0^oo "e"^(-4x) x^4 "d"x`
If f(x) = `{{:(x^2"e"^(-2x)",", x ≥ 0),(0",", "otherwise"):}`, then evaluate `int_0^oo "f"(x) "d"x`
Integrate `((2"a")/sqrt(x) - "b"/x^2 + 3"c"root(3)(x^2))` w.r.t. x
Find `int sqrt(10 - 4x + 4x^2) "d"x`
If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.
If `intx^3/sqrt(1 + x^2) "d"x = "a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"`, then ______.
