Advertisements
Advertisements
Question
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Advertisements
Solution
Let I = `int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Let 7x + 3 = `A[d/dx(3 + 2x - x^2)] + B`
= A(2 – 2x) + B
∴ 7x + 3 = -2Ax + (2A + B)
Comparing the coefficient of x and constant on both the sides, we get
– 2A = 7 and 2A + B = 3
∴ A = `(-7)/(2) and 2(-7/2) + "B" ` = 3
∴ B = 10
∴ 7x + 3 = `(-7)/(2)(2 - 2x) + 10`
∴ I = `int ((-7)/(2)(2 - 2x) + 10)/sqrt(3 + 2x - x^2).dx`
= `(-7)/(2) int ((2 - 2x))/sqrt(3 + 2x - x^2).dx + 10 int(1)/sqrt(3 + 2x - x^2)x`
= `(-7)/(2)"I"_1 + 10"I"_2`
In I1, put 3 + 2x – x2 = t
∴ (2 – 2x)dx = dt
∴ I1 = `int (1)/sqrt(t)dt`
= `int t^(-1/2) dt`
= `t^(1/2)/(1/2) + c_1`
= `2sqrt(3 + 2x - x^2) + c_1`
I2 = `int (1)/sqrt(3 - (x^2 - 2x + 1) + 1).dx`
= `int (1)/sqrt((2)^2 - (x - 1)^2).dx`
= `sin^-1((x - 1)/2) + c_2`
∴ I = `-7sqrt(3 + 2x - x^2) + 10sin^-1((x - 1)/2) + c`, where c = c1 + c2
RELATED QUESTIONS
Evaluate :`intxlogxdx`
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Solve:
dy/dx = cos(x + y)
Write a value of
Write a value of
Write a value of\[\int \log_e x\ dx\].
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Integrate the following w.r.t. x : x3 + x2 – x + 1
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : cos7x
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Evaluate `int (1 + x + x^2/(2!))`dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int x^2/sqrt(1 - x^6)` dx = ________________
`int logx/x "d"x`
`int cos^7 x "d"x`
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
`int dx/(1 + e^-x)` = ______
`int (cos x)/(1 - sin x) "dx" =` ______.
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate `int (1)/(x(x - 1))dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate `int 1/(x(x-1)) dx`
What must be done before integrating after obtaining \[du\]?
What is \[\int\cot t\,dt\] in the evaluation of \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
How should a substitution be chosen?
For indefinite integrals, what should be done after integration in the new variable?
