Question

$\int_{}^{}\frac{\{ x + \sqrt{(1 + x^{2}})\}^{15}}{\sqrt{1 + x^{2}}}$dx = A {x +$\sqrt{1 + x^{2}}$}ⁿ + C, then –

• A. A = $\frac{1}{15}$, n = 15
• B. A = $\frac{1}{14}$, n = 14
• C. A = $\frac{1}{16}$, n = 16
• D. None of these

Answer 1

Answer: (A)

A = $\frac{1}{15}$, n = 15

Answer 2

Let I = $\int_{}^{}\frac{\{ x + \sqrt{(1 + x^{2}})\}^{15}}{\sqrt{1 + x^{2}}}$dx

Here a > 0

Put $\sqrt{(1 + x^{2})}$= t – x (Euler's substitution)

Ž x + $\sqrt{(1 + x^{2})}$ = t \ $\left( 1 + \frac{x}{\sqrt{(1 + x^{2})}} \right)$dx = dt

Ž = dt or $\frac{dx}{\sqrt{(1 + x^{2})}}$ = $\frac{dt}{t}$then I = $\int_{}^{}\frac{t^{15}dt}{t}$

= $\int \mathrm { t } ^ { 14 } \mathrm { dt }$ = $\frac{t^{15}}{15}$ + c

= +c.