What is $f (x) = \int x + 3 x \sqrt{x^{2} + 1} d x$ $f(x) = int x+3xsqrt(x^2+1) dx$ if $f (2) = 7$ $f(2) = 7$ ?

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What is $f (x) = \int x + 3 x \sqrt{x^{2} + 1} d x$ $f(x) = int x+3xsqrt(x^2+1) dx$ if $f (2) = 7$ $f(2) = 7$ ?

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Answer 1 · 2016-07-02T09:22:26

$\Rightarrow f (x) = \frac{x^{2}}{2} + {(x^{2} + 1)}^{\frac{3}{2}} + 5 (1 - \sqrt{5})$ $\implies f(x) = x^2/2+(x^2+1)^{3/2} + 5(1- sqrt5)$

Explanation:

$f (x) = \int x + 3 x \sqrt{x^{2} + 1} d x$ $f(x) = int x+3xsqrt(x^2+1) dx$

noting the pattern $D (α {(x^{2} + 1)}^{\frac{3}{2}}) = α \frac{3}{2} {(x^{2} + 1)}^{\frac{1}{2}} 2 x = 3 α x {(x^{2} + 1)}^{\frac{1}{2}}$ $D ( alpha (x^2+1)^{3/2} ) = alpha 3/2 (x^2+1)^{1/2} 2x = 3 alpha x (x^2+1)^{1/2}$ so here $α = 1$ $alpha = 1$

$\Rightarrow f (x) = \frac{x^{2}}{2} + {(x^{2} + 1)}^{\frac{3}{2}} + C$ $\implies f(x) = x^2/2+(x^2+1)^{3/2} +C$

$7 = 2 + {(5)}^{\frac{3}{2}} + C \Rightarrow C = 5 (1 - \sqrt{5})$ $7= 2+(5)^{3/2} +C implies C = 5(1- sqrt5)$

$\Rightarrow f (x) = \frac{x^{2}}{2} + {(x^{2} + 1)}^{\frac{3}{2}} + 5 (1 - \sqrt{5})$ $\implies f(x) = x^2/2+(x^2+1)^{3/2} + 5(1- sqrt5)$

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What is $f (x) = \int x + 3 x \sqrt{x^{2} + 1} d x$ $f(x) = int x+3xsqrt(x^2+1) dx$ if $f (2) = 7$ $f(2) = 7$ ?

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What is f(x) = int x+3xsqrt(x^2+1) dxf(x)=∫x+3x√x2+1dxf(x) = int x+3xsqrt(x^2+1) dx if f(2) = 7 f(2)=7f(2) = 7 ?

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What is $f (x) = \int x + 3 x \sqrt{x^{2} + 1} d x$ $f(x) = int x+3xsqrt(x^2+1) dx$ if $f (2) = 7$ $f(2) = 7$ ?