Question

4.1 integration
theorem
a constant factor can be moved to the front of an indefinite integral:
$$ int c f ( x ) d x = c int f ( x ) d x. $$
the integral of a sum or difference is the sum or difference of integrals:
$$ int ( f ( x ) pm g ( x ) ) d x = int f ( x ) d x pm int g ( x ) d x. $$
this theorem tells us that multiplied constants stay and the integral can be evaluated term by term.
example:
$$ int ( 3 x ^ { 4 } + 5 x + 6 ) d x = 3 int x ^ { 4 } d x + 5 int x d x + int 6 d x = 3 \frac { x ^ { 5 } } { 5 } + 5 \frac { x ^ { 2 } } { 2 } + 6 x + c = \frac { 3 } { 5 } x ^ { 5 } + \frac { 5 } { 2 } x ^ { 2 } + 6 x + c $$
question 5. evaluate the following integrals.
(a) $$ int ( 6 x ^ { 2 } - 7 x + 3 ) d x $$
(b) $$ int ( 2 x + 4 e ^ { 8 x } ) d x $$
page 173 of 230

Explanation:

Step1: Evaluate the integral of \(6x^{2}\)

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(y = 6x^{2}\), \(\int6x^{2}dx=6\int x^{2}dx=6\times\frac{x^{2+1}}{2 + 1}=2x^{3}\)

Step2: Evaluate the integral of \(-7x\)

For \(y=-7x\), \(\int-7xdx=-7\int xdx=-7\times\frac{x^{1 + 1}}{1+1}=-\frac{7}{2}x^{2}\)

Step3: Evaluate the integral of \(3\)

For \(y = 3\), \(\int3dx=3x\)

Step4: Combine the results

\(\int(6x^{2}-7x + 3)dx=\int6x^{2}dx-\int7xdx+\int3dx=2x^{3}-\frac{7}{2}x^{2}+3x+C\)

Step1: Evaluate the integral of \(2x\)

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(y = 2x\), \(\int2xdx=2\times\frac{x^{1+1}}{1 + 1}=x^{2}\)

Step2: Evaluate the integral of \(4e^{8x}\)

Using the formula \(\int e^{ax}dx=\frac{1}{a}e^{ax}+C\), for \(y = 4e^{8x}\), \(\int4e^{8x}dx=4\times\frac{1}{8}e^{8x}=\frac{1}{2}e^{8x}\)

Step3: Combine the results

\(\int(2x + 4e^{8x})dx=\int2xdx+\int4e^{8x}dx=x^{2}+\frac{1}{2}e^{8x}+C\)

Answer:

\(2x^{3}-\frac{7}{2}x^{2}+3x+C\)