⌠ydx= ⌠(2x⁴-4/√x +3)dx
In integration, we basically add 1 to each power of the given variable and then divide the term by the new power.
Hence 2x⁴ will become 2x⁵. Now we divide 2x⁵ by 5, thus, making it 2x⁵/5.
When solving -4/√x, we will first bring the √x in the numerator, thus the power of x changes from +½ to -½. Now we repeat the same steps. We add one to the power of x, which would transform x^-½ to x^½, which is ultimately √x. Then as we did earlier, we divide √x by ½, which makes it 2√x.
Lastly, if we integrate any constant, we always attach a variable to it with a degree of power of 1. Thus, changing 3 to 3x and since 3x divided by 1 would be 3x, so it remains as it is.
Now if we write these individual terms collectively:
Y = 2x⁵/5 + 4(2√x) + 3x + K
Y = 2x⁵/5 + 8√x + 3x + K
In integration, if the question hasn’t defined a limit for the values of x or whichever variable the equation is in terms of, then you will always add a constant at the end of the equation, which will be denoted by the letter “K”.
