Advertisements
Advertisements
प्रश्न
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
Advertisements
उत्तर
We have I = `int (x^2 + x)/(x^4 - 9) "d"x`
= `int x^2/(x^4 - 9) "d"x + (x"d"x)/(x^4 - 9)`
= I1 + I2
Now I1 = int x^3/(x^4 - 9)`
Put t = x4 – 9
So that 4x3 dx = dt.
Therefore I1 = `1/4 int "dt"/"t"`
= `1/4 log|"t"| + "C"_1`
= `1/4 log|x^4 - 9| + "C"_1`
Again, I2 = `int (x"d"x)/(x^4 - 9)`
Put x2 = u
So that 2x dx = du
Then I2 = `1/2 int "du"/("u"^2 - (3)^2)`
= `1/(2 xx 6) log|("u" - 3)/("u" + 3)| + "C"_2`
= `1/12 log|(x^2 - 3)/(x^2 + 3)| + "C"_2`.
Thus I = I1 + I2
= `1/4 log|x^4 - 9| + 1/12 log|(x^2 - 3)/(x^2 + 3)| + "C"`
APPEARS IN
संबंधित प्रश्न
\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]
Prove that:
\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\] equals
If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\] then the value of I10 + 90I8 is
The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is
Evaluate : \[\int\limits_0^\pi \frac{x}{1 + \sin \alpha \sin x}dx\] .
\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\left( 1 + \cos x \right)^2} dx\]
\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]
\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]
\[\int\limits_0^\pi \frac{x \tan x}{\sec x + \tan x} dx\]
\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]
\[\int\limits_2^3 e^{- x} dx\]
\[\int\limits_1^3 \left( x^2 + 3x \right) dx\]
Using second fundamental theorem, evaluate the following:
`int_0^3 ("e"^x "d"x)/(1 + "e"^x)`
Evaluate the following using properties of definite integral:
`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`
Choose the correct alternative:
Using the factorial representation of the gamma function, which of the following is the solution for the gamma function Γ(n) when n = 8 is
Choose the correct alternative:
Γ(n) is
If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4ex + 5e –x| + C, then ______.
Verify the following:
`int (2x + 3)/(x^2 + 3x) "d"x = log|x^2 + 3x| + "C"`
`int x^9/(4x^2 + 1)^6 "d"x` is equal to ______.
Which integral has Constant C present?
Using \[F\] as an antiderivative, how is \[\int_{a}^{b}f(x)\,dx\] evaluated?
