Question

evaluate the integral.
int_{1}^{4}left(4 x^{2}+5
ight) d x
int_{1}^{4}left(4 x^{2}+5
ight) d x=quad \text { (type an integer or a simplified fraction.) }

Explanation:

Step1: Find the antiderivative

Use the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)).
For \(\int(4x^{2}+5)dx\), we have \(\int4x^{2}dx+\int5dx\).
\(\int4x^{2}dx=4\times\frac{x^{3}}{3}=\frac{4x^{3}}{3}\), \(\int5dx = 5x\).
So the antiderivative \(F(x)=\frac{4x^{3}}{3}+5x\).

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 \(a = 1\), \(b = 4\).
\(F(4)=\frac{4\times4^{3}}{3}+5\times4=\frac{4\times64}{3}+20=\frac{256}{3}+20=\frac{256 + 60}{3}=\frac{316}{3}\).
\(F(1)=\frac{4\times1^{3}}{3}+5\times1=\frac{4}{3}+5=\frac{4 + 15}{3}=\frac{19}{3}\).
Then \(F(4)-F(1)=\frac{316}{3}-\frac{19}{3}\).

Step3: Calculate the result

\(\frac{316-19}{3}=\frac{297}{3}=99\).

Answer:

\(99\)