Advertisements
Advertisements
Question
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Options
log (3x) – log (9x) + c·
log (x) – (log 3) · log (log 9x) + c
log 9 – (log x) · log (log 3x) + c
log (x) + (log 3) · log (log 9x) + c
Advertisements
Solution
log (x) – (log 3) · log (log 9x) + c
[ Hint : `int (log3x)/(xlog(x))*dx = int (log((9x)/3))/(xlog(9x))*dx`
= `int (log (9x) - log3)/(xlog(9x))*dx`
= `int[1/x- (log3)/(xlog(9x))]*dx`
= `int 1/x*dx - (log3) int ((1/x))/(log (9x))*dx`
= log (x) – (log 3) · log (log 9x) + c].
APPEARS IN
RELATED QUESTIONS
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x sin x.
Integrate the function in `x^2e^x`.
Integrate the function in x log x.
Integrate the function in x cos-1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in x (log x)2.
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
`int x^2 e^4x`dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int 1/(4x + 5x^(-11)) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int 1/(x(x - 1)) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int_0^1 x tan^-1 x dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
`intsqrt(1+x) dx` = ______
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`int (logx)^2 dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
