triple integral of x — step-by-step solution

1) Let's compute the integral:

\int x d x

ⓘ An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int x d x

\frac{x^{2}}{2}

\frac{x^{2}}{2}

Indefinite integrals are defined up to an additive constant C, so this yields:

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

2) Let's compute the integral:

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

ⅰ

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

\int \frac{x^{2}}{2} d x + \int C d x

\int \frac{x^{2}}{2} d x + \int C d x

ⓘ A constant factor stays in front of the integral. An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

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

\frac{1}{2} \cdot \frac{x^{3}}{3}

■The derivative of

C x

C

, so

C x

is an antiderivative of

C

\int C d x

C x

\frac{1}{2} \cdot \frac{x^{3}}{3} + C x

ⓘ To multiply fractions, multiply the numerators together and the denominators together:

\frac{1}{2} \cdot \frac{x^{3}}{3}

\frac{1 \cdot x^{3}}{2 \cdot 3}

\frac{x^{3}}{6}

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

Indefinite integrals are defined up to an additive constant C1, so this yields:

\frac{x^{3}}{6} + C x + C1

3) Let's integrate:

\int (\frac{x^{3}}{6} + C x + C1) d x

ⅰ

\int (\frac{x^{3}}{6} + C x + C1) d x

\int \frac{x^{3}}{6} d x + \int C x d x + \int C1 d x

\int \frac{x^{3}}{6} d x + \int C x d x + \int C1 d x

ⓘ A constant factor stays in front of the integral. An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int \frac{x^{3}}{6} d x

\frac{1}{6} \cdot \frac{x^{4}}{4}

\int C x d x

C \cdot \frac{x^{2}}{2}

■The derivative of

C1 \cdot x

C1

, so

C1 \cdot x

is an antiderivative of

C1

\int C1 d x

C1 \cdot x

\frac{1}{6} \cdot \frac{x^{4}}{4} + C \cdot \frac{x^{2}}{2} + C1 \cdot x

ⓘ To multiply fractions, multiply the numerators together and the denominators together:

\frac{1}{6} \cdot \frac{x^{4}}{4}

\frac{1 \cdot x^{4}}{6 \cdot 4}

\frac{x^{4}}{24}

■Multiplying a fraction by a term multiplies its numerator:

C \cdot \frac{x^{2}}{2}

\frac{C x^{2}}{2}

\frac{x^{4}}{24} + \frac{C x^{2}}{2} + C1 \cdot x

▶Indefinite integrals are defined up to an additive constant C2, so this yields:

\frac{x^{4}}{24}

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

+ C1 \cdot x

+ C2

, where C, C1 and C2 are 3 constants