What is the integral of (2x^2)/(sqrt(8+x^2)) ?

The integral of (2x^2)/(sqrt(8+x^2)) is -8ln|x+sqrt(x^2+8)|+12ln(2)+xsqrt(8+x^2)+C

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integral of (2x^2)/(sqrt(8+x^2))

\int \frac{2 x ^{2}}{8 + x ^{2}} d x

- 8 ln x + x^{2} + 8 + 12 ln (2) + x 8 + x^{2} + C

Solution steps

\int \frac{2 x ^{2}}{8 + x ^{2}} d x

Take the constant out:

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

= 2 \cdot \int \frac{x ^{2}}{8 + x ^{2}} d x

Apply Trigonometric Substitution

: \int 8 tan^{2} (u) sec (u) d u

= 2 \cdot \int 8 tan^{2} (u) sec (u) d u

Take the constant out:

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

= 2 \cdot 8 \cdot \int tan^{2} (u) sec (u) d u

Rewrite using trig identities

= 2 \cdot 8 \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Expand

(- 1 + sec^{2} (u)) sec (u) : - sec (u) + sec^{3} (u)

= 2 \cdot 8 \cdot \int - sec (u) + sec^{3} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 \cdot 8 (- \int sec (u) d u + \int sec^{3} (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= 2 \cdot 8 (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arctan (\frac{1}{2 2} x)

= 2 \cdot 8 (- ln tan (arctan (\frac{1}{2 2} x)) + sec (arctan (\frac{1}{2 2} x)) + \frac{1}{2} sec (arctan (\frac{1}{2 2} x)) tan (arctan (\frac{1}{2 2} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{2 2} x)) + sec (arctan (\frac{1}{2 2} x)))

Simplify

2 \cdot 8 (- ln tan (arctan (\frac{1}{2 2} x)) + sec (arctan (\frac{1}{2 2} x)) + \frac{1}{2} sec (arctan (\frac{1}{2 2} x)) tan (arctan (\frac{1}{2 2} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{2 2} x)) + sec (arctan (\frac{1}{2 2} x))) : - 8 ln x + x^{2} + 8 + 12 ln (2) + x 8 + x^{2}

= - 8 ln x + x^{2} + 8 + 12 ln (2) + x 8 + x^{2}

Add a constant to the solution

= - 8 ln x + x^{2} + 8 + 12 ln (2) + x 8 + x^{2} + C

\int \frac{x}{( x ^{2} + x + 1 ) ^{2}} d x

\int \frac{1 + cos ( 4 t )}{2} d t

\int \frac{1}{( 3 x )} d x

cos (t)

\int (arctan (x))^{2} d x

What is the integral of (2x^2)/(sqrt(8+x^2)) ?
The integral of (2x^2)/(sqrt(8+x^2)) is -8ln|x+sqrt(x^2+8)|+12ln(2)+xsqrt(8+x^2)+C