What is the integral of (x^3)/3 sqrt(1+x^4) ?

The integral of (x^3)/3 sqrt(1+x^4) is 1/18 (1+x^4)^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3)/3 sqrt(1+x^4)

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

\frac{1}{18} (1 + x^{4})^{\frac{3}{2}} + C

Solution steps

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

Take the constant out:

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

= \frac{1}{3} \cdot \int x^{3} 1 + x^{4} d x

Apply u-substitution

: \int \frac{u}{4} d u

= \frac{1}{3} \cdot \int \frac{u}{4} d u

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{4} \cdot \int u d u

Apply the Power Rule

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

= \frac{1}{3} \cdot \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + x^{4}

= \frac{1}{3} \cdot \frac{1}{4} \cdot \frac{2}{3} (1 + x^{4})^{\frac{3}{2}}

Simplify

\frac{1}{3} \cdot \frac{1}{4} \cdot \frac{2}{3} (1 + x^{4})^{\frac{3}{2}} : \frac{1}{18} (1 + x^{4})^{\frac{3}{2}}

= \frac{1}{18} (1 + x^{4})^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{18} (1 + x^{4})^{\frac{3}{2}} + C

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

\int (2 x - 9) d x

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

\int 2 - 6 x d x

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

What is the integral of (x^3)/3 sqrt(1+x^4) ?
The integral of (x^3)/3 sqrt(1+x^4) is 1/18 (1+x^4)^{3/2}+C