Solved: "Evaluate the definite integral. ∫14(8x3−x)dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

David Young

Answered question

2021-12-27

Evaluate the definite integral.

\int_{1}^{4} (8 x^{3} - x) d x

Answer & Explanation

eskalopit

Beginner2021-12-28Added 31 answers

Step 1
Given that ,
The integral

\int_{1}^{4} (8 x^{3} - x) d x

By using integration formulas,

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

\int c f (x) d x = c \int f (x) d x

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

Step 2

\int_{1}^{4} (8 x^{3} - x) d x

= \int_{1}^{4} 8 x^{3} d x - \int_{1}^{4} x d x

= 8 \int_{1}^{4} x^{3} d x - \int_{1}^{4} x d x

= 8 {[\frac{x^{4}}{4}]}_{1}^{4} - {[\frac{x^{2}}{2}]}_{1}^{4}

= 2 {[x^{4}]}_{1}^{4} - {[\frac{x^{2}}{2}]}_{1}^{4}

= 2 [4^{4} - 1] - [\frac{4^{2}}{2} - \frac{1}{2}]

= 2 [256 - 1] - [\frac{16}{2} - \frac{1}{2}]

= 510 - [\frac{15}{2}]

= \frac{1005}{2}

Mary Herrera

Beginner2021-12-29Added 37 answers

\int (8 \cdot x^{3} - x) d x

Lets

Vasquez

Expert2022-01-08Added 669 answers

$\begin{matrix} W e h a v e : \\ \int_{1}^{4} 8 x^{3} - x d x \\ \int 8 x^{3} - x d x \\ U s e r u l e s o f i n t e g r a l s . \\ \int 8 x^{3} d x - \int x d x \\ 2 x^{4} - \frac{x^{2}}{2} \\ (2 x^{4} - \frac{x^{2}}{2}) |_{1}^{4} \\ 2 * x^{4} - \frac{4^{2}}{2} - (2 * 1^{4} - \frac{1^{2}}{2}) \\ A n s w e r : \\ \frac{1005}{2} \end{matrix}$

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