Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Solution
Let I = e3logx(x4 + 1)–1.dx
= `int e^(logx^3)/(x^4 + 1).dx`
= `int x^3/(x^4 + 1).dx` ...[∵ elogN = N]
= `(1)/(4) int(4x^3)/(x^4 + 1).dx`
= `(1)/(4) int(d/dx(x^4 + 1))/(x^4 + 1).dx`
= `(1)/(4)log|x^4 + 1| + c. ...[∵ int (f'(x))/f(x) dx = log|f(x)| + c]`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Solve: dy/dx = cos(x + y)
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Evaluate the following.
`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
Evaluate `int (5"x" + 1)^(4/9)` dx
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int "e"^sqrt"x"` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int 1/(cos x - sin x)` dx = _______________
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int (log x)/(log ex)^2` dx = _________
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int x^x (1 + logx) "d"x`
`int 1/(xsin^2(logx)) "d"x`
Choose the correct alternative:
`int(1 - x)^(-2) dx` = ______.
`int sec^6 x tan x "d"x` = ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate `int(1+ x + x^2/(2!)) dx`
Evaluate the following.
`int 1/(x^2+4x-5) dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate `int(1 + x + x^2/(2!))dx`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int sqrt((a - x)/x) dx`
`int "cosec"^4x dx` = ______.
Evaluate `int 1/(x(x-1))dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
