Question

यदि `intx^3/sqrt(1 + x^2) "d"x = "a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"` है, तो ______

Options

• a = `1/3`, b = 1

• a = `(-1)/3`, b = 1

• a = `(-1)/3`, b = –1

• a = `1/3`, b = –1

Answer 1

सही उत्तर `underline("a" = 1/3, "b" = –1)` है।

व्याख्या:

मान लीजिए I = `intx^3/sqrt(1 + x^2) "d"x`

1 + x² = t रखें

⇒ 2x dx = dt

⇒ x dx = `"dt"/2`

∴ I = `1/2 int "t"/sqrt("t") "dt" - 1/2 int 1/sqrt("t") "dt"`

= `1/2 int sqrt("t")  "dt" - 1/2 int "t"^((-1)/2)  "dt"`

= `1/2 xx 2/3 ("t")^(3/2) - 1/2 * 2sqrt("t") + "C"`

= `1/3(1 + x^2)^(3/2) - sqrt(1 + x^2) + "C"`

लेकिन I = `"a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"`

हमें प्राप्त समान पदों की तुलना करने पर,

∴ a = `1/3` और b = –1.