Question

evaluate the integral.
int_{2}^{4}left(4 x^{3}+2
ight) d x
int_{2}^{4}left(4 x^{3}+2
ight) d x=quad \text { (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 \(\int(4x^{3}+2)dx\), we have:
\(\int(4x^{3}+2)dx=4\int x^{3}dx+2\int dx\)
\(=4\times\frac{x^{4}}{4}+2x+C=x^{4}+2x + C\)

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 \(F(x)\) is the antiderivative of \(f(x)\).
Here \(a = 2\), \(b = 4\) and \(F(x)=x^{4}+2x\)
\(F(4)-F(2)=(4^{4}+2\times4)-(2^{4}+2\times2)\)
First, calculate \(4^{4}+2\times4\):
\(4^{4}=256\), \(2\times4 = 8\), so \(4^{4}+2\times4=256 + 8=264\)
Then, calculate \(2^{4}+2\times2\):
\(2^{4}=16\), \(2\times2=4\), so \(2^{4}+2\times2=16 + 4=20\)
Now, \(F(4)-F(2)=264-20\)

Answer:

\(244\)