What is the intx(x+1)^3"d"x?

2 Answers
Jun 30, 2018

intx(x+1)^3dx=1/20(x+1)^4(4x-1)+"c"

Explanation:

To find intx(x+1)^3dx, we use integration by parts

Let u=xrArrdu=dx

and dv=(x+1)^3dxrArrv=1/4(x+1)^4

Thus

intx(x+1)^3dx=1/4x(x+1)^4-int1/4(x+1)^4=

=1/4x(x+1)^4-1/20(x+1)^5+"c"

=1/20(x+1)^4(4x-1)+"c"

Jun 30, 2018

I=x^5/5+(3x^4)/4+(3x^3)/3+x^2/2+c

Explanation:

We know that

color(blue)((1)(a + b)^3 = a^3 + b^3 + 3ab( a + b )

Using (1) we get

I=intx(x+1)^3dx

=>I=intx[x^3+1^3+3x(1)(x+1)]dx

=>I=intx[x^3+1+3x^2+3x]dx

=>I=intx[x^3+3x^2+3x+1]dx

=>I=int[x^4+3x^3+3x^2+x]dx

Using Power Rule:

=>I=x^5/5+(3x^4)/4+(3x^3)/3+x^2/2+c