Advertisements
Advertisements
Question
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Advertisements
Solution
`intsqrt(1 + sin 5x).dx`
= `intsqrt(sin^2 (5x)/2 + cos^2 (5x)/2 + 2sin (5x)/2 cos (5x)/2) dx`
= `intsqrt((cos (5x)/2 + sin (5x)/2)^2) dx`
= `int(cos (5x)/2 + sin (5x)/2) dx`
= `intcos (5x)/2 dx + sin (5x)/2 dx`
= `(sin (5x)/2)/(5/2) - (cos (5x)/2)/(5/2) + "c"`
∴ I = `2/5 (sin (5x)/2-cos (5x)/2) + "c"`
APPEARS IN
RELATED QUESTIONS
Evaluate :`intxlogxdx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Write a value of
Write a value of
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
Evaluate `int (1 + x + x^2/(2!))`dx
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
`int 1/(cos x - sin x)` dx = _______________
`int(log(logx))/x "d"x`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int ((x + 1)(x + log x))^4/(3x) "dx" =`______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Evaluate `int(1+ x + x^2/(2!)) dx`
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate:
`int sqrt((a - x)/x) 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`
If f'(x) = 4x3- 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
Which substitution is appropriate for \[\sqrt{\frac{x}{x-a}}\], \[\sqrt{\frac{x-a}{x}}\], \[\sqrt{x(x-\mathrm{a})}\], or \[\frac{1}{\sqrt{x(x-\mathrm{a})}}\]?
After putting \[t=\cos x\], which integral is obtained from \[\int\sin^2x\cos^2x(\sin x)\,dx\]?
