Integration shortcuts of [f(x)]^n also has derivative f'(x) (Integration Tips)

This integration trick can be applied when f(x) has a power n (here n ≠ 1) And derivate of f(x) is also persent of front of [f(x)]^n. Keep in mind always that n should not be equal to -1

In such case just take the f(x), take its power with plus 1 as power and divide by new power.

Let us we apply this integration rule on following examples.

Integration of x^2 or Integration of x raise to power 2

/ 2
| x dx
/

/ 2
= | x . 1 dx
/

(Here f(x) is x and 2 is its power, derivative of x is 1 which is also exist in front of x. So apply the above rule. Just take f(x), take its power with plus 1 as power and divide by new power.)

2 + 1
x
= ------- + C
2 + 1
3
x
= ----- + C
3

Integration of 1

/
| 1 dx
/

/ 0
= | x . dx
/
/ 0
= | x . 1 dx
/

(Here f(x) is x and 0 is its power, derivative of x is 1 which is also exist in front of x. So apply the above rule. Just take f(x), take its power with plus 1 as power and divide by new power.)

0 + 1
x
= ------- + C
0 + 1
1
x
= ----- + C
1
= x + C

Integration of (ln(x))/x

/
| ln(x)
| ----- dx
| x
/

/
| 1
= | ln(x) . --- dx
| x
/

(Here f(x) is ln(x) and 1 is its power, derivative of ln(x) is 1/x which is also exist in front of ln(x).So apply the above integration rule. Just take f(x), take its power with plus 1 as power and divide by new power.)

1 + 1
(ln(x))
= ------- + C
1 + 1
2
(ln x)
= ----- + C
2

Integration of (2x +3)^(1/2)

/
| 1/2
| (2x + 3) dx
/

/
1 | 1/2
= ---| (2x + 3) . 2 dx
2 |
/

(Here f(x) is 2x + 3 and 1/2 is its power, its derivative was not exit so we try to make its derivative by multiply and divided by 2. Now derivative of 2x + 3 is 2 which is in front of 2x + 3. So apply the above rule. Just take f(x), take its power with plus 1 as power and divide by new power.)