Advertisements
Advertisements
प्रश्न
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Advertisements
उत्तर
Let I = `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Put `"x"."e"^"x" = "t"`
Differentiate w.r.t. x,
`"x" ."e"^"x" +"e"^"x" . 1 = "dt"/"dx" => "e"^"x" ("x" +1)"dx" ="dt"`
`therefore int 1/("cos"^2 "t") "dt"`
= ∫ sec2 t dt
= tan t + c
` therefore int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx" = "tan"("x" ."e"^"x") + "c"`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
`int logx/x "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int cot^2x "d"x`
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
`int ("d"x)/(x(x^4 + 1))` = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
`int 1/(sinx.cos^2x)dx` = ______.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
`int x^2/sqrt(1 - x^6)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`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
Evaluate:
`intsqrt(sec x/2 - 1)dx`
Which expression results after simplifying \[\int\frac{\sin(t-a)}{\sin t}\,dt\]?
