Question

Let $\int_{}^{}{g(x)}dx$ = F(x), then $\int_{}^{}{x^{3}g(x^{2})dx}$ equals to –

• A. $\frac{1}{2}$ [x² (F(x))² – $\int_{}^{}{(F(x))^{2}}$dx]
• B. $\frac{1}{2}$ [x² F(x²) – $\int_{}^{}{(F(x^{2}))}$d (x²)]
• C. $\frac{1}{2}$ [x² F(x) – $\frac { 1 } { 2 }$ $\int_{}^{}{(F(x))^{2}}$dx]
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{2}$ x² F(x²) – $\int_{}^{}{(F(x^{2}))}$d (x²)

Answer 2

$\int x \cdot x ^ { 2 } g \left( x ^ { 2 } \right) d x$ ̃ Let x² = t ̃

=$\frac { 1 } { 2 }$[tF(t)–$\int \mathrm { F } ( \mathrm { t } ) \mathrm { dt }$] = $\frac { 1 } { 2 }$[x² F(x²)–]

$\int x \cdot x ^ { 2 } g \left( x ^ { 2 } \right) d x$ ̃ Let x² = t ̃

=$\frac { 1 } { 2 }$[tF(t)–$\int \mathrm { F } ( \mathrm { t } ) \mathrm { dt }$] = $\frac { 1 } { 2 }$[x² F(x²)–]

$\int x \cdot x ^ { 2 } g \left( x ^ { 2 } \right) d x$ ̃ Let x² = t ̃

=$\frac { 1 } { 2 }$[tF(t)–$\int \mathrm { F } ( \mathrm { t } ) \mathrm { dt }$] = $\frac { 1 } { 2 }$[x² F(x²)–]