Solving Online Antiderivatives

 In calculus, an anti-derivative or unclear integral of a function f is a task F whose imitative is equal to f, i.e., F ′ = f. The opposite operation of differentiation is called Anti-derivatives. The solving process for anti-derivatives is called anti-differentiation and which is the development of result  derived. The anti-derivative is also known as integral calculus. 

                      If F(x) is an function and derivative of that function with respect to x is f(x) , then we say that the integral of f(x) with respect  to x is F(x).  `int f(x) dx = F(x) `

 

Solving Anti-derivative formulas online:

 

1. ` int` x n dx = `(x^n+1) / (n+1)`

2.` int ` cos x .dx = sin x. + c

3. `int` sin x.dx = - cos x + c

4. `int` sec2x. dx = tan x + c

5.` int` cosec x.cot x. dx = -cosec x + c

6. `int` sec x.tan x .dx = sec x + c

7. `int ` cosec2 x .dx = -cot x + c

8.  `int` `(1/x) ` dx = log x + c

9.  `int` e x dx = e x + c

10. `int` u dv = uv -`int` v du

  

Antiderivative formula for inverse trigonometric terms:

1. `int (dx) / (x^2 + a^2)` =` (1/a) tan^-1(x / a) + c`

2. `int (dx) / (a^2 - x^2) ` = `(1/(2a)) log [(a + x) / (a - x)] + c`

3. `int (dx) / (x^2 - a^2)` = `(1/(2a)) log [(x - a) / (x + a)] + c`

4. `int (dx) / sqrt(a^2 - x^2)` = `sin^-1(x / a) + c`

5. `int (dx) / sqrt(x^2 - a^2) ` = `log [(x + sqrt(x^2 - a^2)] + c`

6. `int (dx) / sqrt(x^2 + a^2) ` = `log [(x + sqrt(x^2 + a^2)] + c`

 

Solving Examples for Anti-derivatives online:

 

Solving online anti-derivative problem 1:

            Integrate the given inverse trigonometric function: x tan-1 x

   Solution:

                         `int` tan-1x dx

               Take u = tan-1x                       dv = x dx

                       `du = [1/(1+ x^2)]dx`                v = `x^2/2`

                        `int` tan-1x dx   = uv - `int ` v du

                                             =  `x^2/2` tan-1 x   -  `int ` `x^2/2`` [1/(1+ x^2)]dx`   

                                             =  `x^2/2` tan-1 x - `(1/2)` ` int`   `[x^2 / (1+ x^2)]dx`

                                             =  `x^2/2`  tan-1 x -` (1/2)` `int`  `(x^2 + 1 - 1) / (1+ x^2)dx`  

                                             =   `x^2/2` tan-1 x - `(1/2)` `int`  `[1 - [1/ (1+ x^2)]] dx`

                                             =    `x^2/2` tan-1 x - `(1/2)`   [x - tan-1 x] + c

       Answer:

              `int` tan-1 x dx   =   `x^2/2` tan-1 x  - `(1/2)` [ x - tan-1 x] + c

 

Solving online anti-derivative problem 2:

  Evaluate:` int` (x2+5)1/3x dx

      Solution:

               Let u = x2+5

         Therefore,` (du)/(dx)` = 2x        or    `(dx)/(du)` = `1/(2x)`

   Now,`int` (x2+5)1/3x dx =`int` (x2+5)1/3x ( `(dx)/(du)`) du

                                         =`int` (u1/3 × x × ( `1/(2x)`) du

                                         = `(1/2)` `int` u1/3du

                                         = `(1/2)`  × `(3/4)` u4/3

                                         =` (3/8)` (x2+ 5)4/3 + c

     Answer:

            ` int` (x2+5)1/3x dx   = `(3/8)` (x2+5)4/3 + c

Solving online anti-derivative problem 3:

2. Integrate:   `int` cos42x dx

    Solution:

           `int`cos42x dx = `int` (cos22x)2dx

                               = `int` `((1+ cos 4x) / 2)^2dx`

                               = `(1/4) ` `int` ( 1+2 cos 4x + cos24x) dx

                               =` (1/4)` `int`` ( 1+2 cos 4x + (( 1+ cos 8x ) /2)) dx`

                               =` (1/8)` `int` (2 + 4 cos 4x + 1+ cos 8x) dx

                               = `(1/8)` `int` (3 + 4 cos 4x + cos 8x) dx

                               = `(1/8)`` [3x + ((4 sin 4x) / 4) + ((sin 8x ) / 8)] + c`

                               = `(1/64) ` [24x + 8 sin 4x + sin 8x] +c

    Answer:

          `int`cos42x dx = `(1/64)` [24x + 8 sin 4x + sin 8x] +c

 

Solving online anti-derivative practice problems:

Solving online anti-derivative practice problem 1:

    Integrate: `int` `(1/ (sin x + cos x))` dx

    Answer: ` (1/ sqrt2)` ` log [sec (x-(pi/4)) + tan (x-(pi/4))] + c`

Solving online anti-derivative practice problem 2:

  Integrate `int` (x10   + 2x4) dx

Answer: `(1/11)` x11   `(2/5)` x5     dx