Advertisements
Advertisements
Question
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Solution
`Let I= int(x-3)sqrt(x^2+3x-18x)dx`
`Put sqrt(x^2+3x−18)=t ⇒(x^2+3x−18) =t^2`
On differentiating with respect to x, we get:
`2x+3=2t(dt/dx)`
`x+3/2=t(dt/dx)`
`x+3/2+3−3=t(dt/dx)`
`x−3+9/2=t(dt/dx)..............(1)`
The given integral can be rewritten as follows:
`I=int(x−3+9/2-9/2)sqrt(x^2+3x-18)dx`
`=int(x-3+9/2)sqrt(x^2+3x+18)dx-9/2intsqrt(x^2+3x+18)dx..............(2)`
Suppose that `l_1=int(x-3+9/2)sqrt(x^2_3x-18)dx`
`"On using equation "(1), we getl_1=intt^2dt=t^3/3+C_1=(x^2+3x-18)^(3/2)/3+C_1`
Suppose that `l_2=intsqrt(x^2+3x-18)dx`
`intsqrt(x^2+3x-18)dx=intsqrt((x+3/2)^2-(9/2)^2)dx`
`=((2x+3)/4) sqrt(x^2+3x-18)-81/8log|(2x+3)/2+sqrt(x^2+3x-18)|+C_2`
`l=(x^2+3x-18)^(3/2)/3-9/8(2x+3)sqrt(x^2+3x-18)+729/16log|(2x+3)/2+sqrt(x^2+3x-18)|+C`
where C=C_1+C_2 is a constant.
APPEARS IN
RELATED QUESTIONS
Find `intsqrtx/sqrt(a^3-x^3)dx`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Write a value of
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
`int x^3"e"^(x^2) "d"x`
`int (cos x)/(1 - sin x) "dx" =` ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate `int (1+x+x^2/(2!)) dx`
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate the following.
`int 1/ (x^2 + 4x - 5) 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).
