Advertisements
Advertisements
Question
Solve: `int sqrt(4x^2 + 5)dx`
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 : sec3 x w. r. t. x.
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Integrate the function in x2 log x.
Integrate the function in x tan-1 x.
Integrate the function in x (log x)2.
Integrate the function in `e^x (1/x - 1/x^2)`.
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int (sinx)/(1 + sin x) "d"x`
`int ("d"x)/(x - x^2)` = ______
∫ log x · (log x + 2) dx = ?
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Evaluate:
`int (logx)^2 dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
