Advertisements
Advertisements
Question
Verify the following:
`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`
Advertisements
Solution
L.H.S. = `int (2x - 1)/(2x + 3) "d"x`
⇒ `int (1 - 4/(2x + 3)) "d"x` .....[Dividing the numerator by the denominator]
⇒ `int 1 * "d"x - 4 int 1/(2x + 3) "d"x`
⇒ `int 1 * "d"x - 4/2 int 1/(x + 3/2) "d"x`
⇒ `int 1 * "d"x - 2 int 1/(x + 3/2) "d"x`
⇒ `x - 2 log |x + 3/2| + "C"`
⇒ `x - 2 log |(2x + 3)/2| + "C"`
⇒ `x - log|((2x + 3)/2)^2| + "C"` ....[∵ n log m = log mn]
⇒ `x - log |(2x + 3)^2| - log 2^2 + "C"`
⇒ `x - log |(2x + 3)^2| + "C"_1`
⇒ R.H.S. ......[Where C1 = C – log 22]
L.H.S. = R.H.S.
Hence proved.
APPEARS IN
RELATED QUESTIONS
Evaluate the following definite integrals:
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
Evaluate each of the following integral:
Evaluate each of the following integral:
Evaluate each of the following integral:
If \[\int_0^a \frac{1}{4 + x^2}dx = \frac{\pi}{8}\] , find the value of a.
The derivative of \[f\left( x \right) = \int\limits_{x^2}^{x^3} \frac{1}{\log_e t} dt, \left( x > 0 \right),\] is
Evaluate : \[\int e^{2x} \cdot \sin \left( 3x + 1 \right) dx\] .
\[\int\limits_0^1 \left| 2x - 1 \right| dx\]
\[\int\limits_0^\pi \frac{x}{a^2 \cos^2 x + b^2 \sin^2 x} dx\]
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx\]
\[\int\limits_0^{\pi/2} \frac{\cos^2 x}{\sin x + \cos x} dx\]
\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx\]
\[\int\limits_0^{\pi/2} \frac{1}{2 \cos x + 4 \sin x} dx\]
\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]
Using second fundamental theorem, evaluate the following:
`int_0^1 "e"^(2x) "d"x`
Evaluate the following:
Γ(4)
Evaluate the following:
`int_0^oo "e"^(-mx) x^6 "d"x`
Evaluate the following integrals as the limit of the sum:
`int_1^3 x "d"x`
Choose the correct alternative:
`int_0^1 (2x + 1) "d"x` is
Choose the correct alternative:
Γ(n) is
Choose the correct alternative:
If n > 0, then Γ(n) is
Choose the correct alternative:
`int_0^oo x^4"e"^-x "d"x` is
If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.
Evaluate the following:
`int ((x^2 + 2))/(x + 1) "d"x`
Find: `int logx/(1 + log x)^2 dx`
