Question

3⁵ˣ⁻²+ 1/(2x+1)³,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

Dear Student,

Answer:
$= \frac{ {3}^{5x} }{45 \: log \: 3} - \frac{1}{4( {2x + 1)}^{2} } + c$

Solution:

Simplify the function ,and apply linearity

3⁵ˣ⁻²+ 1/(2x+1)³
${3}^{(5x - 2)} + \frac{1}{ {(2x + 1)}^{3} } \\ \\ = {3}^{5x} {3}^{ - 2} + \frac{1}{ {(2x + 1)}^{3} } \\ = \frac{ {3}^{5x} }{9} + \frac{1}{ {(2x + 1)}^{3} }$

As we know that integration of
${a}^{x} dx = \frac{ {a}^{x} }{log(a)}$
So , integration of first term
$\frac{1}{9} \frac{( {3}^{5x} )}{5 \: log3} \\ ..........eq1$
for second part ,let 2x+1 = t

2dx = dt

dx = dt/2

$= \frac{1}{2} \frac{1}{ {t}^{3} } dt \\ \\ = - \frac{1}{4} ( \frac{1}{ {t}^{2} } ) + c........eq2$
for complete result add both 1 and 2

$= \frac{ {3}^{5x} }{45 \: log \: 3} - \frac{1}{4( {2x + 1)}^{2} } + c$

or
$= \frac{ {3}^{5x-2} }{5 \: log \: 3} - \frac{1}{4( {2x + 1)}^{2} } + c$


is the final answer.

Hope it helps you.