Advertisements
Advertisements
प्रश्न
Integrate the functions:
sin (ax + b) cos (ax + b)
Advertisements
उत्तर
Let `I = int sin (ax + b) cos (ax + b) dx`
Put sin (ax + b) = t
⇒ a cos (ax + b) dx = dt
∴ `I = 1/a int t dt = 1/a * t^2/2 + C`
`= 1/(2a) t^2 + C`
`= 1/ (2a) sin^2 (ax + b) + C`
Or, put cos (ax + b) = t
⇒ -a sin (ax + b) dx = dt
∴ `I = (-1)/a int dt = (-1)/a t^2/2 + C`
`= (-cos^2 (ax + b))/(2a) + C`
APPEARS IN
संबंधित प्रश्न
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Evaluate `int 1/((2"x" + 3))` dx
Evaluate: `int "x" * "e"^"2x"` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int(sin2x)/(5sin^2x+3cos^2x) dx=` ______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
Evaluate `int1/(x(x - 1))dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
What is integration by substitution?
Which standard substitution is used for \[\sqrt{x^2+\mathrm{a}^2}\], \[\frac{1}{\sqrt{x^2+\mathrm{a}^2}}\], or \[x^2+\mathrm{a}^2\]?
What is the value of \[\int\sin^3x\cos^2x\,dx\]?
For \[\int\frac{\sin x}{\sin(x+a)}\,dx\], which substitution gives \[dx=dt\]?
Which expression results after simplifying \[\int\frac{\sin(t-a)}{\sin t}\,dt\]?
