Direct integrals for the polynomials

The direct integrals are the simplest type of integral that exists, since they are solved straight away by applying what we know, which is essentially that integrating is the opposite of deriving.

These are the integrals of functions of the type k⋅xn, where k is a constant and n is any number other than −1.

We have then :∫k⋅xn dx=k⋅xn+1n+1+C note that if we take the derivative we obtain: ddx(k⋅xn+1n+1+C)=ddx(k⋅xn+1n+1)+ddxC=k⋅ddx(xn+1n+1)= =k⋅(n+1)xnn+1=k⋅xn

Example

∫x dx=x22+C, since ddx(x22+C)=x

Example

∫x2 dx=x33+C, since ddx(x33+C)=x2

Example

∫x3 dx=x44+C, since ddx(x44+C)=x3

Example

∫23x5 dx=23x66+C, since ddx(23x66+C)=23⋅ddx(x66)=23⋅6⋅x6−16=23⋅x5

Example

∫x dx=∫x12=x12+112+1+C=x3232+C=23x32, since ddx(23x32+C)=ddx(23x32)+ddxC=23⋅ddxx23=23⋅32x12=x12=x

Example

∫4253⋅x23 dx=4253⋅x5353+C, since ddx(4253⋅x5353+C)=ddx(4253⋅x5353)+ddx(C)=4253⋅ddx(x5353)=

=4253⋅x23

Example

∫15x−2 dx=15x−1−1+C=−151x+C, since ddx(−151x)=−15⋅ddx(1x)=−15⋅x−2−1=15⋅x−2