Question

please solve this question​

Answer 1

Given Integral :-

$\red{\leadsto}\:$$\sf \large\int \: (2x+1)(x+4)^5 dx$

Solution :-

$\begin{gathered}\\\implies\quad \sf \int \: (2x+1)(x+4)^5 \\\end{gathered}$

Substitute x bt t-4 -

$\sf x= t- 4$

Differentiating both sides-

$\quad\sf \dfrac{dx}{dt} = 1$

$\quad\sf dx= dt$

$\begin{gathered}\\\implies\quad \sf \int \:(2(t-4)+1)(t-4+4)^5 \:dt\\\end{gathered}$

$\begin{gathered}\\\implies\quad \sf \int \:(2t-8+1)(t)^5 \:dt\\\end{gathered}$

$\begin{gathered}\\\implies\quad \sf \int \: (2t-7)t^5 \:dt\\\end{gathered}$

$\begin{gathered}\\\implies\quad \sf \int \: 2t^6-7t^5 \:dt\\\end{gathered}$

$\begin{gathered}\\\implies\quad \sf \int \: 2\frac{t^7}{7}- 7\frac{t^6}{6} + C \\\end{gathered}$

$\begin{gathered}\\\implies\quad \sf \int \: \frac{2t^7}{7}- \frac{7t^6}{6} +C \\\end{gathered}$

Now, x= t-4

so , t = x+4

Replacing t by x+4 ,

$\begin{gathered}\\\implies\quad \sf \int \: \frac{2(x+4)^7}{7}- \frac{7(x+4)^6}{6} +C \\\end{gathered}$