Question

Integrate
y
=
x
3
+
3
x
2
+
2
x
+
4

Answer 1

To integrate the function \( y = x^3 + 3x^2 + 2x + 4 \), we will integrate each term of the polynomial separately with respect to \( x \).

The integral of a polynomial term \( ax^n \) is given by:

\[
\int ax^n \, dx = \frac{a}{n+1} x^{n+1} + C
\]

where \( C \) is the constant of integration.

Now, we can apply this to each term:

1. For \( x^3 \):
\[
\int x^3 \, dx = \frac{1}{4} x^4
\]

2. For \( 3x^2 \):
\[
\int 3x^2 \, dx = 3 \cdot \frac{1}{3} x^3 = x^3
\]

3. For \( 2x \):
\[
\int 2x \, dx = 2 \cdot \frac{1}{2} x^2 = x^2
\]

4. For \( 4 \):
\[
\int 4 \, dx = 4x
\]

Now, combine all the results:

\[
\int (x^3 + 3x^2 + 2x + 4) \, dx = \frac{1}{4} x^4 + x^3 + x^2 + 4x + C
\]

Thus, the result of the integral is:

\[
\int (x^3 + 3x^2 + 2x + 4) \, dx = \frac{1}{4} x^4 + x^3 + x^2 + 4x + C
\]