If f≤ft(t+1/2t+1)=t+1 then f(x) \, dx =

Correct Answer is : x2+14log⁡(2x−1)+cx/2+1/4 (2x-1)+c2x+41log(2x−1)+c

Correct Answer is : x2+14log⁡(2x−1)+cx/2+1/4 (2x-1)+c2x+41log(2x−1)+c

Test Index

TS EAMCET 4 May 2018 Shift 2 Solved Paper

Section: Mathematics

Question : 72 of 160

Marks: +1, -0

f\left(\frac{t+1}{2t+1}\right)=t+1

\int f(x) \, dx =

Solution:

Consider the expression,

f\left(\frac{t+1}{2t+1}\right)=t+1

Take,

\frac{t+2}{2t+1}=x

t=\frac{1-x}{2x-1}

This implies,

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

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

f(x)=\frac{x}{2x-1}

So,

\int f(x) \, dx = \int \frac{x}{2x-1} \, dx

=\frac{1}{2} \int \frac{2x}{2x-1} \, dx

=\frac{1}{2} \int \frac{2x-1+1}{2x-1} \, dx

=\frac{1}{2} \left[ x+\frac{1}{2} \log(2x-1) \right] + c

Further simplify the above,

\int f(x) \, dx = \frac{x}{2}+\frac{1}{4} \log(2x-1)+c

Go to Question:

Prev Question

Next Question