Question

Evaluate the following:
1. [tex] {\displaystyle \rm \int (4x+6)² dx
2. [tex] {\displaystyle \rm \int_{0}^{2} (4x+6)² dx } [\tex]

Answer 1

To Evaluate:

$1){\displaystyle \rm \int (4x+6)² dx }$

$2){\displaystyle \rm \int_{0}^{2} (4x+6)² dx }$

Solution :

$1){\displaystyle \rm \int (4x+6)² dx }$

$\implies {\displaystyle \rm \int(16x {}^{2} + 36 + 48x)  dx }$

Now, as we know that :

$\to\displaystyle \rm  \int (f(x) + g(x) + c)dx$

$\displaystyle \rm =   \int f(x) dx+  \int g(x)dx + \int cdx$

So, we get that :

$\implies {\displaystyle \rm \int(16x {}^{2} + 36 + 48x)  dx }$

${ \implies \displaystyle \rm \int16x {}^{2} dx+ \int 48xdx +  \int36  dx }$

Now, using the Formula :

$\bull \displaystyle \rm \int {x}^{n} dx =  \frac{ {x}^{n  + 1} }{n + 1}$

Where n ≠ -1

So, now Evaluating :

${ \implies\displaystyle \rm \int  16 {x}^{2} dx  +  \int48x dx+  \int36dx}$

${\implies \displaystyle \rm 16 \int  {x}^{2} dx + 48 \int xdx  +  \int36dx}$

${ \implies \displaystyle \rm  \left(16 \times  \frac{ {x}^{2 + 1} }{2 + 1}  \right) +  \left(48 \times  \frac{ {x}^{1 + 1} }{1 + 1}  \right) + 36x + c}$

${ \implies \displaystyle \rm  \left(16 \times  \frac{ {x}^{3} }{3}  \right) +  \left(48 \times  \frac{ {x}^{2} }{2}  \right) + 36x + c}$

${ \implies \displaystyle \rm \left(\frac{16 {x}^{3} }{3}  \right) +  \left(   {24x}^{2}  \right) + 36x + c}$

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$2){\displaystyle \rm \int_{0}^{2} (4x+6)² dx }$

 

Now, for solving this definite integral, we will first apply the value of the upper limit then subtract the value applied from the lower limit  

${ \implies\displaystyle \rm \int_{0}^{2} (4x+6)² dx }$

$\implies \displaystyle \rm \left(\frac{16x^3}{3} + 24x^2+36x\right)\bigg|_{0}^2$

$\implies \displaystyle \rm \left(\frac{16\cdot2^3}{3} +24\cdot2^3+36\cdot2\right)-\left(\frac{16\cdot0^3}{3} +24\cdot0^2+36\cdot0\right)$

$\implies \displaystyle \rm \frac{128}{3} +96+72-0$

$\implies \displaystyle \rm 42\frac{2}{3} +96+72$

$\implies \displaystyle \rm 210\frac{2}{3}$

Learn More:

$\boxed{\begin{array}{c|c}\bf f(x)&\bf\displaystyle\int\rm f(x)\:dx\\ \\ \frac{\qquad\qquad}{}&\frac{\qquad\qquad}{}\\ \rm k&\rm kx+C\\ \\ \rm sin(x)&\rm-cos(x)+C\\ \\ \rm cos(x)&\rm sin(x)+C\\ \\ \rm{sec}^{2}(x)&\rm tan(x)+C\\ \\ \rm{cosec}^{2}(x)&\rm-cot(x)+C\\ \\ \rm sec(x)\ tan(x)&\rm sec(x)+C\\ \\ \rm cosec(x)\ cot(x)&\rm-cosec(x)+C\\ \\ \rm tan(x)&\rm log(sec(x))+C\\ \\ \rm\dfrac{1}{x}&\rm log(x)+C\\ \\ \rm{e}^{x}&\rm{e}^{x}+C\\ \\ \rm x^{n},n\neq-1&\rm\dfrac{x^{n+1}}{n+1}+C\end{array}}$