Math shortcuts, Articles, worksheets, Exam tips, Question, Answers, FSc, BSc, MSc

#### Keep Connect with Us

• =

• Welcome in Math School.
• This is beta verion of our website.
##### 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.)

```                      1
---  + 1
2
1  (2x + 3)
=  --- --------  + C
2   1
--- + 1
2

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

3/2
(2x + 3)
=  --------
3

```

##### SHORTCUT TRICKS (Division)
• Divisible by 2 Shortcut trick
• Divisible by 3 Shortcut trick
• Divisible by 4 Shortcut trick
• Divisible by 5 Shortcut trick
• Divisible by 6 Shortcut trick
• Divisible by 7 Shortcut trick
• Divisible by 8 Shortcut trick
• Divisible by 9 Shortcut trick
• Divisible by 10 Shortcut trick