How to Solve Anti Derivatives

In mathematics, anti derivative is also called as integration process. Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function f. In this case it is called an indefinite integral. Integral can be classified as definite and indefinite integral. (Source: Wikipedia)

General formula for integration:

∫ `(x^n)` = `(x^(n + 1) / (n + 1))` + c

Example problems for how to solve anti derivatives

Anti derivative example problem 1:

Solve:

Integrate the given function ∫ (14.3x4 + 31.7x3 - 6x) dx.

Solution:

Given ∫ (14.3x4 + 31.7x3 - 6x) dx

Integrate the given function with respect to x, we get

∫ (14.3x4 + 31.7x3 - 6x) dx = ∫ 14.3x4 dx + ∫ 31.7x3 dx - ∫ 6x dx.

= 14.3 (x5 / 5) + 31.7 (x4 / 4) - 6 (x2 / 2) + c.

= `(14.3 / 5)` x5 + `(31.7 / 4)` x4 - 3 x2 + c.

Answer:

The final answer is  `(14.3 / 5)` x5 + `(31.7 / 4)` x4 - 3 x2 + c.

Anti derivative example problem 2`:`

Solve:

Find the value of the integration

`int_0^3(x^10)dx`

Solution:

Integrate the given function with respect to x, we get

`int_0^3(x^10)dx`  = `(x^11 / 11)`30

Substitute the lower and upper limits, we get

= `((3^11 / 11) - (0^11/ 11))`

= `((177147 / 11) - (0 / 11))`

= `(177147 / 11)`

Answer:

The final answer is `(177147 / 11)`

Anti derivative example problem 3:

Solve:

Integration using algebraic rational function ∫ `(15 dx) / (8x + 21)`

Solution:

Using integrable function method,

Given function is ∫ `(15dx) / (8x + 21)`

Formula:

∫ [L / (ax + c)] dx = (L / a) log (ax + c)

From given, L = 15, a = 8, and c = 21

Integrate the given equation with respect to x, we get

= `(15 / 8)` log (8x + 21)

Answer:

The final answer is `(15 / 8)` log (8x + 21).

Practice problems for how to solve anti derivatives

Anti derivative practice problem 1:

Solve:

Integrate the given function using integrable function ∫ `(2 / (12x + 103))` dx

Answer:

The final answer is `(1 / 6)` log (12x +103)

Anti derivative practice problem 2:

Solve:

Integrate the given function using integrable function ∫ `(27 / (21x + 48))` dx

Answer:

The final answer is `(9 / 7)` log (21x + 48)

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Anti derivative practice problem 3:

Solve:

Integrate the given function ∫ (27.39x3 - 17.132x) dx

Answer:

The final answer is `(27.39 / 4)` x4 - `(17.132 / 2)` x2 + c