Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Advertisements
Solution
Let I = `int sqrt((x - 3)(7 - x)).dx`
=`intsqrt(-x^2 + 10x - 21).dx`
= `int sqrt(- (x^2 - 10x + 21)).dx`
= `int sqrt(4 -(x^2 - 10x + 25)).dx`
= `int sqrt(2^2 - (x - 5)^2`
= `((x - 5)/2) sqrt(2^2 - (x - 5)^2) + 2^2/(2) sin^-1 ((x - 5)/2) + c`
= `((x - 5)/2) sqrt((x - 3)(7 - x)) + 2sin^-1 ((x - 5)/2) + 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`
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 cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in e2x sin x.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int 1/x "d"x` = ______ + c
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`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int 1/sqrt(x^2 - a^2)dx` = ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate `int(1 + x + x^2/(2!))dx`.
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
