# Math Snap

## $\int_{3}^{6} 2 \sqrt{x}+\frac{6}{x^{3}}+10$

#### STEP 1

Assumptions 1. We are given a definite integral to evaluate. 2. The integral is from $x = 3$ to $x = 6$. 3. The integrand is $2 \sqrt{x} + \frac{6}{x^3} + 10$.

#### STEP 2

Write down the integral in its given form. $\int_{3}^{6} \left( 2 \sqrt{x} + \frac{6}{x^3} + 10 \right) \, dx$

#### STEP 3

Separate the integral into three simpler integrals. $\int_{3}^{6} 2 \sqrt{x} \, dx + \int_{3}^{6} \frac{6}{x^3} \, dx + \int_{3}^{6} 10 \, dx$

#### STEP 4

Evaluate the first integral $\int_{3}^{6} 2 \sqrt{x} \, dx$.

#### STEP 5

Rewrite $2 \sqrt{x}$ as $2 x^{1/2}$. $\int_{3}^{6} 2 x^{1/2} \, dx$

#### STEP 6

Use the power rule for integration $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. $\int 2 x^{1/2} \, dx = 2 \cdot \frac{x^{3/2}}{3/2} = \frac{4}{3} x^{3/2}$

#### STEP 7

Evaluate the definite integral from 3 to 6. $\left[ \frac{4}{3} x^{3/2} \right]_{3}^{6}$

#### STEP 8

Plug in the upper limit $x = 6$. $\frac{4}{3} (6)^{3/2}$

#### STEP 9

Calculate $6^{3/2}$. $6^{3/2} = (6^1)^{3/2} = 6^{1.5} = 6 \sqrt{6}$

#### STEP 10

Simplify $\frac{4}{3} \cdot 6 \sqrt{6}$. $\frac{4}{3} \cdot 6 \sqrt{6} = 8 \sqrt{6}$

#### STEP 11

Plug in the lower limit $x = 3$. $\frac{4}{3} (3)^{3/2}$

#### STEP 12

Calculate $3^{3/2}$. $3^{3/2} = (3^1)^{3/2} = 3^{1.5} = 3 \sqrt{3}$

#### STEP 13

Simplify $\frac{4}{3} \cdot 3 \sqrt{3}$. $\frac{4}{3} \cdot 3 \sqrt{3} = 4 \sqrt{3}$

#### STEP 14

Subtract the lower limit result from the upper limit result. $8 \sqrt{6} - 4 \sqrt{3}$

#### STEP 15

Evaluate the second integral $\int_{3}^{6} \frac{6}{x^3} \, dx$.

#### STEP 16

Rewrite $\frac{6}{x^3}$ as $6 x^{-3}$. $\int_{3}^{6} 6 x^{-3} \, dx$

#### STEP 17

Use the power rule for integration. $\int 6 x^{-3} \, dx = 6 \cdot \frac{x^{-2}}{-2} = -3 x^{-2}$

#### STEP 18

Evaluate the definite integral from 3 to 6. $\left[ -3 x^{-2} \right]_{3}^{6}$

#### STEP 19

Plug in the upper limit $x = 6$. $-3 (6)^{-2}$

#### STEP 20

Calculate $6^{-2}$. $6^{-2} = \frac{1}{6^2} = \frac{1}{36}$

#### STEP 21

Simplify $-3 \cdot \frac{1}{36}$. $-\frac{3}{36} = -\frac{1}{12}$

#### STEP 22

Plug in the lower limit $x = 3$. $-3 (3)^{-2}$

#### STEP 23

Calculate $3^{-2}$. $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

#### STEP 24

Simplify $-3 \cdot \frac{1}{9}$. $-\frac{3}{9} = -\frac{1}{3}$

#### STEP 25

Subtract the lower limit result from the upper limit result. $-\frac{1}{12} - (-\frac{1}{3}) = -\frac{1}{12} + \frac{1}{3} = \frac{1}{4}$

#### STEP 26

Evaluate the third integral $\int_{3}^{6} 10 \, dx$.

#### STEP 27

Since 10 is a constant, use the rule $\int a \, dx = a x + C$. $\int_{3}^{6} 10 \, dx = 10 x \bigg|_{3}^{6}$

#### STEP 28

Evaluate the definite integral from 3 to 6. $10 (6) - 10 (3) = 60 - 30 = 30$

#### STEP 29

Combine the results of the three integrals. $8 \sqrt{6} - 4 \sqrt{3} + \frac{1}{4} + 30$

##### SOLUTION

The final result is: $8 \sqrt{6} - 4 \sqrt{3} + \frac{1}{4} + 30$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 51, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://studdy.ai/shared-solution/b3172316-bdbe-4eef-a2da-b1e607debc90", "fetch_time": 1718640676000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-26/segments/1718198861733.59/warc/CC-MAIN-20240617153913-20240617183913-00716.warc.gz", "warc_record_offset": 496011237, "warc_record_length": 18483, "token_count": 1311, "char_count": 2981, "score": 4.6875, "int_score": 5, "crawl": "CC-MAIN-2024-26", "snapshot_type": "latest", "language": "en", "language_score": 0.4539734721183777, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00018-of-00064.parquet"}