Advertisements
Advertisements
Question
Evaluate the following: `int x.sin^-1 x.dx`
Advertisements
Solution
Let I = `int x.sin^-1 x.dx`
= `int (sin^-1x).xdx`
= `(sin^-1x) int x.dx - int[{d/dx(sin^-1x) int x.dx}].dx`
= `(sin^-1x) (x^2/2) - int (1/sqrt(1 - x^2))(x^2/2).dx`
= `(x^2 sin^-1x)/(2) + (1)/(2) int (-x^2)/sqrt(1 - x^2).dx`
= `(x^2 sin^-1x)/(2) + (1)/(2) int ((1 - x^2) - 1)/sqrt(1 - x^2).dx`
= `(x^2 sin^-1x)/(2) + (1)/(2) int [sqrt(1 - x^2) - (1)/sqrt(1 - x^2)].dx`
= `(x^2 sin^-1x)/(2) + (1)/(2) int sqrt(1 - x^2).dx - (1)/(2) int (1)/sqrt(1 - x^2).dx`
= `(x^2 sin^-1x)/(2) + (1)/(2) [x/2 sqrt(1 - x^2) + 1/2 sin^-1x] - 1/2 sin^-1 x + c`
= `(x^2 sin^-1x)/(2) + (1)/(4) xsqrt(1 - x^2) - (1)/(4) sin^-1 x + c`.
APPEARS IN
RELATED QUESTIONS
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Integrate the function in x sin 3x.
Integrate the function in (sin-1x)2.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Find :
`∫(log x)^2 dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
`int "e"^x [x (log x)^2 + 2 log x] "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.
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
`int_0^1 x tan^-1 x dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`int(1-x)^-2 dx` = ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int1/(x+sqrt(x)) dx` = ______
`int logx dx = x(1+logx)+c`
Evaluate:
`inte^x sinx dx`
Evaluate:
`int (logx)^2 dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
The value of `inta^x.e^x dx` equals
Evaluate the following.
`intx^3 e^(x^2)dx`
Which function is an example under priority \(I\) in the LIATE rule?
Which pair is listed as Trigonometric in the LIATE rule?
Which pair is listed as Exponential in the LIATE rule?
For \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx,\] which functions are chosen as the first function and the second function?
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
Repeated parts may be needed for which pair of integrals?
