Question

$\int f^{\prime }(ax+b)\left[f\right(ax+b){]}^{2}dx=$

• A. $f(ax+b)+c$
• B. $\frac{\left[f\right(ax+b){]}^2}{2a}+c$
• C. $\frac{\left[f\right(ax+b){]}^3}{3a}+c$
• D. None of these

Answer 1

Answer: (C)

$\frac{\left[f\right(ax+b){]}^3}{3a}+c$

Now,

$\int f^{\prime }(ax+b)\left[f\right(ax+b){]}^{2}dx$

$=\frac{1}{a}\int f^{\prime }(ax+b)\left[f\right(ax+b){]}^{2}adx$

$=\frac{1}{a}\int \left[f\right(ax+b\left){]}^{2}d\right(f(ax+b))$

$=\frac{1}{a}\frac{\left[f\right(ax+b){]}^3}{3}+c$. [ Where $c$ is integrating constant]