∫ e x [ x 3 + x + 1 ( 1 + x 2 ) 3 / 2 ] d x is equal to Options: A . &nbspx2ex(1+x2)1/2+c B . &nbspexx(1+x2)1/2+c C . &nbspex(1+x2)1/2+c D . &nbspxex(1+x2)1/2+c

Answer: Option D : D I = ∫ e x [ x √ 1 + x 2 + 1 ( 1 + x 2 ) 3 / 2 ] d x Let f ( x ) = x √ 1 + x 2 ⇒ f ′ ( x ) = √ 1 + x 2 − x 2 √ 1 + x 2 ( 1 + x 2 ) = 1 ( 1 + x 2 ) 3 / 2 ∴ I = e x f ( x ) + c = e x x √ 1 + x 2 + c

∫ex[x3+x+1(1+x2)3/2]dx is equal to

Question

\int e^{x} [\frac{x^{3} + x + 1}{(1 + x^{2})^{3 / 2}}] d x

is equal to

Options:

A . x2ex(1+x2)1/2+c

B . exx(1+x2)1/2+c

C . ex(1+x2)1/2+c

D . xex(1+x2)1/2+c

Answer: Option D
:
D

I = \int e^{x} [\frac{x}{\sqrt{1 + x^{2}}} + \frac{1}{(1 + x^{2})^{3 / 2}}] d x

Let

f (x) = \frac{x}{\sqrt{1 + x^{2}}} \Rightarrow f^{^{'}} (x) = \frac{\sqrt{1 + x^{2}} - \frac{x^{2}}{\sqrt{1 + x^{2}}}}{(1 + x^{2})}

= \frac{1}{(1 + x^{2})^{3 / 2}}

∴ I = e^{x} f (x) + c

= \frac{e^{x} x}{\sqrt{1 + x^{2}}} + c

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