What is the integral of 3x^2(x^3+1) ?

The integral of 3x^2(x^3+1) is (x^6)/2+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 3x^2(x^3+1)

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

\frac{x ^{6}}{2} + x^{3} + C

Solution steps

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

Take the constant out:

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

= 3 \cdot \int x^{2} (x^{3} + 1) d x

Expand

x^{2} (x^{3} + 1) : x^{5} + x^{2}

= 3 \cdot \int x^{5} + x^{2} d x

Apply the Sum Rule:

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

= 3 (\int x^{5} d x + \int x^{2} d x)

\int x^{5} d x = \frac{x ^{6}}{6}

\int x^{2} d x = \frac{x ^{3}}{3}

= 3 (\frac{x ^{6}}{6} + \frac{x ^{3}}{3})

Simplify

3 (\frac{x ^{6}}{6} + \frac{x ^{3}}{3}) : \frac{x ^{6}}{2} + x^{3}

= \frac{x ^{6}}{2} + x^{3}

Add a constant to the solution

= \frac{x ^{6}}{2} + x^{3} + C

\int \frac{x ^{3}}{x ^{5} + 7 x ^{3}} d x

\int \frac{x + 1}{16 - x ^{2}} d x

\int sin^{8} (\frac{x}{8}) cos (\frac{x}{8}) d x

\int t^{3} d t

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

What is the integral of 3x^2(x^3+1) ?
The integral of 3x^2(x^3+1) is (x^6)/2+x^3+C