e^{x}≤ft({x+5}{(x+6)²}) d x is

Correct Answer is : exx+6+c{e^{x}}{x+6}+cx+6ex+c, where c is the constant of integration.

{e^{x}}{(x+6)²}+c, where c is the constant of integration.

{e^{x}}{x+5}+c, where c is the constant of integration.

{e^{x}}{(x+5)²}+c, where c is the constant of integration.

Correct Answer is : exx+6+c{e^{x}}{x+6}+cx+6ex+c, where c is the constant of integration.

Test Index

MHTCET 25 Apr 2025 Shift 1 Paper

Question : 114 of 150

Marks: +1, -0

\int e^{x}\left(\frac{x+5}{(x+6)^{2}}\right) d x

Solution:

Concept:

Use the standard result:

\int e^x [f(x) + f'(x)]\,dx = e^x f(x) + c

Explanation:

Rewrite the numerator as

x+5 = (x+6) - 1

Then the integral becomes

\int e^x \left( \frac{x+6-1}{(x+6)^2} \right) dx

Split the fraction:

\int e^x \left( \frac{1}{x+6} - \frac{1}{(x+6)^2} \right) dx

Let

f(x) = \frac{1}{x+6}

. Then

f'(x) = -\frac{1}{(x+6)^2}

So the integrand is

e^x [f(x) + f'(x)]

Applying the formula gives

e^x f(x) + c

Hence,

\int e^x \left( \frac{x+5}{(x+6)^2} \right) dx = \frac{e^x}{x+6} + c

Answer:

\frac{e^x}{x+6} + c

, where

c

is the constant of integration. Therefore, the correct option is D.

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