{1}{{1+4x²} - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.4
Integrate the functions in Exercises 86 to 108.

Question : 87 of 271

Marks: +1, -0

\frac{1}{\sqrt{1+4x^2}}

Solution:

Let

I=\frac{1}{\sqrt{1+4x^2}}dx

=\frac{1}{2}\int\frac{dx}{\sqrt{\frac{1}{4}+x^2}}

=\frac{1}{2}\int\frac{dx}{\sqrt{\left(\frac{1}{2}\right)^2+x^2}}

=\frac{1}{2}\log\left|x+\sqrt{\frac{1}{4}+x^2}\right|+C_1\left[\because\int\frac{dx}{\sqrt{x^2+a^2}}=\log\left|x+\sqrt{x^2+a^2}\right|+C\right]

=\frac{1}{2}\log\left|\frac{2x+\sqrt{1+4x^2}}{2}\right|+C_1

=\frac{1}{2}\log\left|2x+\sqrt{1+4x^2}\right|-\frac{1}{2}\log2+C_1

=\frac{1}{2}\log\left|2x+\sqrt{1+4x^2}\right|+C

\left[C=\frac{-1}{2}\log2+C_1\right]

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