Question

Solve: $\int (1+2x+3x^2+4x^3+......)dx$ for $(|x|<1)$.

• A. $(1+x{)}^{-1}+c$
• B. $(1-x{)}^{-1}+c$
• C. $(1+x{)}^{-2}+c$
• D. None of these

Answer 1

Answer: (B)

$(1-x{)}^{-1}+c$

Let $I=\int (1+2x+3x^2+4x^3+......)dx$

The terms given under the integral are $1+2x+3x^2+....$

The terms in the series are the derivatives of

$x+x^2+x^3+.....$ and consider this to be $L$ for instance.

and its integration will again give us the same thing i.e. $x+x^2+x^3+.....$

So, we get $L$ as the result whose sum is upto infinite terms

$\therefore I=(x+x^2+x^3+x^4+.....)+c=(1-x{)}^{-1}+c$