What is the integral of 2(2+x^2)^{3/2}x ?

The integral of 2(2+x^2)^{3/2}x is 2/5 (2+x^2)^{5/2}+C

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integral of 2(2+x^2)^{3/2}x

\int 2 (2 + x^{2})^{\frac{3}{2}} x d x

\frac{2}{5} (2 + x^{2})^{\frac{5}{2}} + C

Solution steps

\int 2 (2 + x^{2})^{\frac{3}{2}} x d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int (2 + x^{2})^{\frac{3}{2}} x d x

Apply u-substitution

: \int \frac{u ^{\frac{3}{2}}}{2} d u

= 2 \cdot \int \frac{u ^{\frac{3}{2}}}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int u^{\frac{3}{2}} d u

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= 2 \cdot \frac{1}{2} \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = 2 + x^{2}

= 2 \cdot \frac{1}{2} \cdot \frac{2}{5} (2 + x^{2})^{\frac{5}{2}}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{2}{5} (2 + x^{2})^{\frac{5}{2}} : \frac{2}{5} (2 + x^{2})^{\frac{5}{2}}

= \frac{2}{5} (2 + x^{2})^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} (2 + x^{2})^{\frac{5}{2}} + C

(5 - 2) (58)

\int \frac{2 x}{y} d x

y = 7 e^{x} cos (x)

sin (3 x)

(8.17) (3.17)

What is the integral of 2(2+x^2)^{3/2}x ?
The integral of 2(2+x^2)^{3/2}x is 2/5 (2+x^2)^{5/2}+C