What is the integral of sqrt(x+x^4) ?

The integral of sqrt(x+x^4) is 1/3 (x^{3/2}sqrt(1+x^3)+ln|x^{3/2}+sqrt(1+x^3)|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(x+x^4)

\int x + x^{4} d x

\frac{1}{3} (x^{\frac{3}{2}} 1 + x^{3} + ln x^{\frac{3}{2}} + 1 + x^{3}) + C

Solution steps

\int x + x^{4} d x

Factor

x + x^{4} : x (1 + x^{3})

= \int x (1 + x^{3}) d x

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

x (1 + x^{3}) = x (1 + x^{3})

= \int x 1 + x^{3} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{3 u ^{2} - 1} d u

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

Take the constant out:

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

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

Apply Trigonometric Substitution

: \int sec^{3} (v) d v

= \frac{2}{3} \cdot \int sec^{3} (v) d v

Apply Integral Reduction

= \frac{2}{3} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

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

= \frac{2}{3} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Substitute back

= \frac{2}{3} (\frac{sec ^{2} ( arcsec ( 1 + x ^{3} ) ) sin ( arcsec ( 1 + x ^{3} ) )}{2} + \frac{1}{2} ln tan (arcsec (1 + x^{3})) + sec (arcsec (1 + x^{3})))

Simplify

\frac{2}{3} (\frac{sec ^{2} ( arcsec ( 1 + x ^{3} ) ) sin ( arcsec ( 1 + x ^{3} ) )}{2} + \frac{1}{2} ln tan (arcsec (1 + x^{3})) + sec (arcsec (1 + x^{3}))) : \frac{1}{3} (x^{\frac{3}{2}} 1 + x^{3} + ln x^{\frac{3}{2}} + 1 + x^{3})

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

Add a constant to the solution

= \frac{1}{3} (x^{\frac{3}{2}} 1 + x^{3} + ln x^{\frac{3}{2}} + 1 + x^{3}) + C

\int tan^{2} (3 x - 1) sec^{2} (3 x - 1) d x

\int x e^{(x^{2} - 1)} d x

\int \frac{4}{( 1 + 2 x ^{2} ) ^{3}} d x

\int \frac{4 x ^{2} + x + 17}{( x + 1 ) ( x ^{2} + 4 )} d x

\int (x - 1) (2 - x) d x

What is the integral of sqrt(x+x^4) ?
The integral of sqrt(x+x^4) is 1/3 (x^{3/2}sqrt(1+x^3)+ln|x^{3/2}+sqrt(1+x^3)|)+C