Exercise
∫((1+x)4x3)dx
Step-by-step Solution
1
Simplify (1+x)4 using the power of a power property: (am)n=am⋅n. In the expression, m equals 21 and n equals 4
∫(1+x)2x3dx
Intermediate steps
∫1+2x+x2x3dx
3
Divide x3 by 1+2x+x2
;x2+2x;+1;;x;−2;;;x2+2x;+1);x3−;xn−;xn−;xn;x2+2x;+1;−x3−2x2−x;−;xn−x3−2x2−x;;−2x2−x;−;xn;x2+2x;+1−;xn;;2x2+4x;+2;;;;2x2+4x;+2;;−;xn;;3x;+2;;
Intermediate steps
∫(x−2+1+x3+(1+x)2−1)dx
Intermediate steps
5
Simplify the expression
∫xdx+∫−2dx+∫1+x3dx+∫(1+x)2−1dx
Intermediate steps
6
The integral ∫xdx results in: 21x2
Intermediate steps
7
The integral ∫−2dx results in: −2x
Intermediate steps
8
The integral ∫1+x3dx results in: 3ln(x+1)
3ln(x+1)
Intermediate steps
9
The integral ∫(1+x)2−1dx results in: 1+x1
10
Gather the results of all integrals
21x2−2x+3ln∣x+1∣+1+x1
11
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration C
21x2−2x+3ln∣x+1∣+1+x1+C0
Final answer to the exercise
21x2−2x+3ln∣x+1∣+1+x1+C0