integral from 0 to 6 of 1/sqrt(x^2+64)

1) Let's integrate:

\int_{0}^{6} \frac{1}{\sqrt{x^{2} + 64}} d x

The derivative of

\ln (x + \sqrt{x^{2} + 64})

\frac{1 + \frac{x}{\sqrt{x^{2} + 64}}}{x + \sqrt{x^{2} + 64}}

\frac{\frac{\sqrt{x^{2} + 64} + x}{\sqrt{x^{2} + 64}}}{x + \sqrt{x^{2} + 64}}

\frac{1}{\sqrt{x^{2} + 64}}

ⓘ The derivative of

\ln (x + \sqrt{x^{2} + 64})

\frac{1}{\sqrt{x^{2} + 64}}

, so

\ln (x + \sqrt{x^{2} + 64})

is an antiderivative of

\frac{1}{\sqrt{x^{2} + 64}}

\int \frac{1}{\sqrt{x^{2} + 64}} d x

\ln (x + \sqrt{x^{2} + 64})

▷

\ln (x + \sqrt{x^{2} + 64})

2) Let's evaluate over the interval

[0, 6]

\ln (6 + \sqrt{6^{2} + 64}) - \ln (0 + \sqrt{0^{2} + 64})

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

6^{2}

6 \cdot 6

36

■Adding 0 changes nothing:

0 + 64

64

0 + \sqrt{64}

\sqrt{64}

■The square root of a number is the positive number whose square is that number:

\sqrt{64}

8

;

8^{2}

64

\ln (6 + \sqrt{36 + 64}) - \ln (8)

ⓘ Parentheses are computed first:

36 + 64

100

■The logarithm of a power is a multiple of the logarithm, ln(a^n) = n*ln(a):

- \ln (8)

- \ln (2^{3})

- 3 \cdot \ln (2)

\ln (6 + \sqrt{100}) - 3 \cdot \ln (2)

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{100}

10

;

10^{2}

100

\ln (6 + 10) - 3 \cdot \ln (2)

ⓘ Start from

6

and move forward

10

on the number line:

6 + 10

16

\ln (16) - 3 \cdot \ln (2)

ⓘ For strictly positive arguments, a sum of logarithms is the logarithm of the product, a difference that of the quotient, ln(a) + ln(b) = ln(a*b):

\ln (16) - 3 \cdot \ln (2)

\ln (\frac{16}{2^{3}})

\ln (\frac{16}{2^{3}})

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{3}

2 \cdot 2 \cdot 2

8

\ln (\frac{16}{8})

ⓘ We simplify the fraction by dividing the numerator and the denominator by

8

, their greatest common divisor:

\frac{16}{8}

2

▶Integral:

\ln (2)

3) The numeric value of:

\ln (2)

equals:

▷

\approx 0.6931