Advertisements
Advertisements
प्रश्न
Solve: `int sqrt(4x^2 + 5)dx`
उत्तर
`int sqrt(4x^2 + 5).dx = int sqrt((x^2 + 5/4)).dx`
= `2int sqrt(x^2 + 5/4).dx`
= `2int sqrt(x^2 + (sqrt(5)/2)^2).dx`
= `2[x/2 sqrt(x^2 + 5/4) + (5/4)/2 log|x + sqrt(x^2 + 5/4)|] + c_1`
∵ `int sqrt(x^2 + a^2).dx = x/2 sqrt(x^2 + a^2) + a^2/2 log|x + sqrt(x^2 + a^2)| + c`
= `xsqrt(x^2 + 5/4) + 5/4 log|x + sqrt(x^2 + 5/4)| + c_1`
= `x/2 sqrt(4x^2 + 5) + 5/4 log|x + sqrt((4x^2 + 5)/2)| + c_1`
= `x/2 sqrt(4x^2 + 5) + 5/4 log|(2x + sqrt(4x^2 + 5))/2| + c_1`
= `x/2 sqrt(4x^2 + 5) + 5/4 log|2x + sqrt(4x^2 + 5)| - 5/4 log 2 + c`
= `x/2 sqrt(4x^2 + 5) + 5/4 log|2x + sqrt(4x^2 + 5)| + c_1`
Where c = c1 – `5/4` log2, a constant.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x cos-1 x.
Integrate the function in tan-1 x.
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in e2x sin x.
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t.x : log (x2 + 1)
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
Evaluate: ∫ (log x)2 dx
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int tan^-1 sqrt(x) "d"x` is equal to ______.
`int 1/sqrt(x^2 - 9) dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
