$\smallint \,\,\frac{{{4^{x + 1}}\,\, - \,\,{7^{x - 1}}}}{{{{28}^x}}}\;\,\,dx$ is equal to

$\frac{{{4^{ - x}}}}{{{{\log }_e}\,\,7}}\,\, - \,\,\frac{{{7^{ - x}}}}{{{{\log }_e}\,\,4}}\,\, + \,\,c$

$\frac{{{4^{ - x}}}}{{{{\log }_e}\,\,4}}\,\, - \,\,\frac{{{7^{ - x}}}}{{{{\log }_e}\,\,7}}\,\, + \,\,c$

If a function $f\left( x \right)$ satisfies $f'\left( x \right) = g\left( x \right)$ Then, the value of $\mathop \smallint \nolimits_a^b \,\,f\left( x \right)\,\,g\,\,\left( x \right)\,\,dx$ is

$\frac{1}{2}\,\,{\left[ {\,f\left( b \right)\,\, - \,\,f\left( a \right)} \right]^2}$

The value of $\int \log (1-\sqrt{x}) \mathrm{d} x$ is

$x \log (1-\sqrt{x})-\frac{1}{2} x-\frac{3}{2} \sqrt{x}+C$

$x^2 \log (1-\sqrt{x})-\frac{1}{2} \sqrt{x}+C$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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991 Medium

991 of 995

Analyzing +4 | -1

\int \frac{\tan x}{1+\tan x+\tan ^2 x} d x=x-\frac{K}{\sqrt{A}} \tan ^{-1}\left(\frac{K \tan x+1}{\sqrt{A}}\right)+C,

(

C

is a constant of integration), then the ordered pair

(K, A)

is equal to

View answer and explanation

Correct option: A

(2,3)

Option A ((2,3)) is correct because after solving the integral and comparing with the given form, we find K=2 and A=3.

992 Easy

992 of 995

Application +4 | -1

\int \,\,\frac{{{4^{x + 1}}\,\, - \,\,{7^{x - 1}}}}{{{{28}^x}}}\;\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{1}{{7\;\,\,{{\log }_e}4}}\,\,{4^{ - x}}\,\, - \,\,\frac{4}{{{{\log }_e}7}}\,\,{7^{ - x}}\,\, + \,\,c

Option A (

\frac{1}{{7\;\,\,{\log }_e4}}\,{4^{ - x}}\, - \,\frac{4}{{{\log }_e7}}\,{7^{ - x}}\, + \,c

) is correct because it is the result of integrating the given expression after simplification.

993 Easy

993 of 995

Analyzing +4 | -1

\int \sin ^5 x \cos ^4 x \,d x=A \cos ^9 x+B \cos ^7 x+C \cos ^5 x+D,

then

9 A+7 B+5 C=

View answer and explanation

Correct option: B

Option B is correct because after evaluating the integral and comparing coefficients, we find that

9A + 7B + 5C = 0

994 Easy

994 of 995

Application +4 | -1

If a function

f\left( x \right)

satisfies

f'\left( x \right) = g\left( x \right)

Then, the value of

\int \nolimits_a^b \,\,f\left( x \right)\,\,g\,\,\left( x \right)\,\,dx

View answer and explanation

Correct option: A

\frac{1}{2}\,\,\left[ {{{\left( {\,f\left( b \right)} \right)}^2}\,\, - \,\,{{\left( {\,F\left( a \right)} \right)}^2}} \right]

Option A is correct because the integral of f(x)g(x) from a to b, where f'(x) = g(x), evaluates to (1/2)[(f(b))² - (f(a))²].

995 Easy

995 of 995

Application +4 | -1

The value of

\int \log (1-\sqrt{x}) \mathrm{d} x

View answer and explanation

Correct option: B

(x-1) \log (1-\sqrt{x})-\frac{1}{2} x-\sqrt{x}+C

Option B is correct because after integrating by parts and simplifying, the value of k is found to be

\frac{5 \pi}{4}

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