What is the integral from-2 to 2 of x^2sqrt(1+4x^2) ?

The integral from-2 to 2 of x^2sqrt(1+4x^2) is (ln(-4+sqrt(17))-ln(4+sqrt(17))+264sqrt(17))/(64)

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integral from-2 to 2 of x^2sqrt(1+4x^2)

\int_{- 2}^{2} x^{2} 1 + 4 x^{2} d x

\frac{ln ( - 4 + 17 ) - ln ( 4 + 17 ) + 264 17}{64}

Decimal

16.94235 \dots

Solution steps

\int_{- 2}^{2} x^{2} 1 + 4 x^{2} d x

Apply Trigonometric Substitution

= \int_{- a r c t a n (4)}^{a r c t a n (4)} \frac{1}{8} sec^{3} (u) tan^{2} (u) d u

Take the constant out:

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

= \frac{1}{8} \cdot \int_{- a r c t a n (4)}^{a r c t a n (4)} sec^{3} (u) tan^{2} (u) d u

Rewrite using trig identities

= \frac{1}{8} \cdot \int_{- a r c t a n (4)}^{a r c t a n (4)} sec^{3} (u) (- 1 + sec^{2} (u)) d u

Expand

sec^{3} (u) (- 1 + sec^{2} (u)) : - sec^{3} (u) + sec^{5} (u)

= \frac{1}{8} \cdot \int_{- a r c t a n (4)}^{a r c t a n (4)} - sec^{3} (u) + sec^{5} (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

= \frac{1}{8} (- \int_{- a r c t a n (4)}^{a r c t a n (4)} sec^{3} (u) d u + \int_{- a r c t a n (4)}^{a r c t a n (4)} sec^{5} (u) d u)

\int_{- a r c t a n (4)}^{a r c t a n (4)} sec^{3} (u) d u = 417 + \frac{1}{2} (ln (4 + 17) - ln (- 4 + 17))

\int_{- a r c t a n (4)}^{a r c t a n (4)} sec^{5} (u) d u = 3717 + \frac{3}{8} ln (4 + 17) - \frac{3}{8} ln (- 4 + 17)

= \frac{1}{8} (- (417 + \frac{1}{2} (ln (4 + 17) - ln (- 4 + 17))) + 3717 + \frac{3}{8} ln (4 + 17) - \frac{3}{8} ln (- 4 + 17))

Simplify

\frac{1}{8} (- (417 + \frac{1}{2} (ln (4 + 17) - ln (- 4 + 17))) + 3717 + \frac{3}{8} ln (4 + 17) - \frac{3}{8} ln (- 4 + 17)) : \frac{ln ( - 4 + 17 ) - ln ( 4 + 17 ) + 264 17}{64}

= \frac{ln ( - 4 + 17 ) - ln ( 4 + 17 ) + 264 17}{64}

\int 5 sin (6 π x) d x

ln ((5 x + 2) (3 x - 4)^{2})

y^{^{''}} - 4 y^{^{'}} + 4 y = x^{2}

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

y^{^{'}} = 6 x (y - 1)^{\frac{2}{3}}

What is the integral from-2 to 2 of x^2sqrt(1+4x^2) ?
The integral from-2 to 2 of x^2sqrt(1+4x^2) is (ln(-4+sqrt(17))-ln(4+sqrt(17))+264sqrt(17))/(64)