What is the integral from 0 to x of sqrt(x^4+x^6) ?

The integral from 0 to x of sqrt(x^4+x^6) is

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integral from 0 to x of sqrt(x^4+x^6)

\int_{0}^{x} x^{4} + x^{6} d x

- \frac{x x ^{2} + 1 + ln x + x ^{2} + 1}{2} + \frac{x ( x ^{2} + 1 ) ^{\frac{3}{2}}}{4} + \frac{3 ( x x ^{2} + 1 + ln x + x ^{2} + 1 )}{8}

Solution steps

\int_{0}^{x} x^{4} + x^{6} d x

Simplify

x^{4} + x^{6} : x^{2} 1 + x^{2}

= \int_{0}^{x} x^{2} 1 + x^{2} d x

Apply Trigonometric Substitution

: \int_{0}^{a r c t a n (x)} sec^{3} (u) tan^{2} (u) d u

= \int_{0}^{a r c t a n (x)} sec^{3} (u) tan^{2} (u) d u

Rewrite using trig identities

= \int_{0}^{a r c t a n (x)} sec^{3} (u) (- 1 + sec^{2} (u)) d u

Expand

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

= \int_{0}^{a r c t a n (x)} - 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

= - \int_{0}^{a r c t a n (x)} sec^{3} (u) d u + \int_{0}^{a r c t a n (x)} sec^{5} (u) d u

\int_{0}^{a r c t a n (x)} sec^{3} (u) d u = \frac{x x ^{2} + 1 + ln x + x ^{2} + 1}{2}

\int_{0}^{a r c t a n (x)} sec^{5} (u) d u = \frac{x ( x ^{2} + 1 ) ^{\frac{3}{2}}}{4} + \frac{3 ( x x ^{2} + 1 + ln x + x ^{2} + 1 )}{8}

= - \frac{x x ^{2} + 1 + ln x + x ^{2} + 1}{2} + \frac{x ( x ^{2} + 1 ) ^{\frac{3}{2}}}{4} + \frac{3 ( x x ^{2} + 1 + ln x + x ^{2} + 1 )}{8}

\int_{0}^{1} x^{3} + x d x

\int_{0}^{a r c t a n (3)} sin (2 x) d x

\int_{0}^{\infty} \frac{3}{x + 5} d x

\int_{- 4}^{4} - e^{\frac{1}{4} x} d x

\int_{0}^{\frac{π}{8}} sin (x) d x

What is the integral from 0 to x of sqrt(x^4+x^6) ?
The integral from 0 to x of sqrt(x^4+x^6) is