Advertisements
Advertisements
Question
Solve: `int sqrt(4x^2 + 5)dx`
Advertisements
Solution
`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
RELATED QUESTIONS
Integrate the function in x2 log x.
Integrate the function in x sin−1 x.
Integrate the function in x cos-1 x.
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^2.log x.dx`
Evaluate the following:
`int x tan^-1 x . dx`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/x "d"x` = ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
Evaluate `int 1/(x(x - 1)) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
`int 1/sqrt(x^2 - a^2)dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`int e^(logcosx)dx`
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
