Advertisements
Advertisements
Question
Evaluate `int (3"a"x)/("b"^2 + "c"^2x^2) "d"x`
Advertisements
Solution
Let v = b2 + c2x2, then dv = 2c2 xdx
Therefore, `int (3"a"x)/("b"^2 + "c"^2x^2) "d"x`
= `(3"a")/(2"c"^2) int "dv"/"v"`
= `(3"a")/("c"^2) log |"b"^2 + "c"^2x^2| + "C"`
APPEARS IN
RELATED QUESTIONS
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
If `f` is an integrable function such that f(2a − x) = f(x), then prove that
If f (x) is a continuous function defined on [0, 2a]. Then, prove that
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
\[\int\limits_0^{\pi/2} \frac{1}{2 + \cos x} dx\] 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: \[\int\limits_{- \pi/2}^{\pi/2} \frac{\cos x}{1 + e^x}dx\] .
Evaluate : \[\int\frac{dx}{\sin^2 x \cos^2 x}\] .
\[\int\limits_0^1 \tan^{- 1} x dx\]
\[\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]
\[\int\limits_{- a}^a \frac{x e^{x^2}}{1 + x^2} dx\]
\[\int\limits_0^\pi x \sin x \cos^4 x dx\]
\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]
\[\int\limits_0^{\pi/2} \frac{\cos^2 x}{\sin x + \cos x} dx\]
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 integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
Evaluate the following integrals as the limit of the sum:
`int_0^1 x^2 "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`
Evaluate the following:
`int ((x^2 + 2))/(x + 1) "d"x`
`int x^9/(4x^2 + 1)^6 "d"x` is equal to ______.
What is the result of a definite integral?
Which integral has Constant C present?
If velocity changes with time, what does a definite integral give over a time interval?
