Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Advertisements
Solution
Let I = `int (1 + x)/(x.sin (x + log x)).dx`
= `int (1)/(sin(x + logx)).((1 + x)/x).dx`
= `int (1)/(sin(x + log x)).(1/x + 1).dx`
Put x + log x = t
∴ `(1 + 1/x).dx` = dt
∴ I = `int (1)/sint dt = int "cosec" t dt`
= log |cosec t – cot t| + c
= log |cosec(x + log x) – cot(x + logx)|+ c.
RELATED QUESTIONS
Find `intsqrtx/sqrt(a^3-x^3)dx`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals:
`int (7x + 3)/sqrt(3 + 2x - x^2).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
`int logx/(log ex)^2*dx` = ______.
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate: `int "x" * "e"^"2x"` dx
Evaluate: `int log ("x"^2 + "x")` dx
`int logx/x "d"x`
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int dx/(1 + e^-x)` = ______
`int(5x + 2)/(3x - 4) dx` = ______
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate:
`int sin^2(x/2)dx`
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int 1/(x(x-1))dx`
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).
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
Integration by substitution is the reverse process of which rule?
