Question

evaluate the integral.

$$ \\int_{2}^{3}\\left(3 x^{3}+2\ ight) d x $$

$$ \\int_{2}^{3}\\left(3 x^{3}+2\ ight) d x=\\square $$

(type an integer or a simplified fraction.)

Explanation:

Step1: Find the antiderivative

Using the power rule for integration \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)) and \(\int a dx=ax + C\) (where \(a\) is a constant).
For \(\int(3x^{3}+2)dx\), we have \(\int3x^{3}dx+\int2dx\).
\(\int3x^{3}dx=3\times\frac{x^{4}}{4}=\frac{3}{4}x^{4}\), \(\int2dx = 2x\). So the antiderivative \(F(x)=\frac{3}{4}x^{4}+2x\).

Step2: Apply the fundamental theorem of calculus

The fundamental theorem of calculus states that \(\int_{a}^{b}f(x)dx=F(b)-F(a)\), where \(a = 2\), \(b = 3\) and \(F(x)=\frac{3}{4}x^{4}+2x\).
\(F(3)=\frac{3}{4}\times3^{4}+2\times3=\frac{3}{4}\times81 + 6=\frac{243}{4}+6=\frac{243 + 24}{4}=\frac{267}{4}\).
\(F(2)=\frac{3}{4}\times2^{4}+2\times2=\frac{3}{4}\times16+4=12 + 4=16=\frac{64}{4}\).
Then \(F(3)-F(2)=\frac{267}{4}-\frac{64}{4}\).

Step3: Calculate the result

\(\frac{267-64}{4}=\frac{203}{4}\).

Answer:

\(\frac{203}{4}\)