How do you integrate $\int 5^{x} - 3^{x} d x$ $int 5^x-3^xdx$ from $[0, 1]$ $[0,1]$ ?

Question

How do you integrate $\int 5^{x} - 3^{x} d x$ $int 5^x-3^xdx$ from $[0, 1]$ $[0,1]$ ?

2 Answers

Impact of this question

3343 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-05-05T15:24:30

The answer is $= 0.66$ $=0.66$

Explanation:

Let $y = 5^{x}$ $y=5^x$

Taking log on both sides

$ln y = x ln 5$ $lny=xln5$

$y = e^{x ln 5}$ $y=e^(xln5)$

Therefore,

$\int 5^{x} d x = \int e^{x ln 5} d x = \frac{e^{x ln 5}}{ln 5} = \frac{5^{x}}{ln 5}$ $int5^xdx=inte^(xln5)dx=e^(xln5)/ln5=5^x/ln5$

Similarly,

$y = 3^{x}$ $y=3^x$

Taking log on both sides

$ln y = x ln 3$ $lny=xln3$

$y = e^{x ln 3}$ $y=e^(xln3)$

Therefore,

$\int 3^{x} d x = \int e^{x ln 3} d x = \frac{e^{x ln 3}}{ln 3} = \frac{3^{x}}{ln 3}$ $int3^xdx=inte^(xln3)dx=e^(xln3)/ln3=3^x/ln3$

Therefore,

$\int_{0}^{1} (5^{x} - 3^{x}) d x = \int_{0}^{1} 5^{x} d x - \int_{0}^{1} 3^{x} d x$ $int_0^1(5^x-3^x)dx=int_0^1 5^xdx-int_0^1 3^xdx$

$= {[\frac{5^{x}}{ln 5} - \frac{3^{x}}{ln 3}]}_{0}^{1}$ $=[5^x/ln5-3^x/ln3]_0^1$

$= (\frac{5}{ln 5} - \frac{3}{ln 3}) - (\frac{1}{ln 5} - \frac{1}{ln 3})$ $=(5/ln5-3/ln3)-(1/ln5-1/ln3)$

$= \frac{4}{ln 5} - \frac{2}{ln 3}$ $=4/ln5-2/ln3$

$= 0.66$ $=0.66$

Answer 2 · 2017-05-05T15:32:18

The integral has value $\frac{4}{ln (5)} - \frac{2}{ln (3)}$ $4/ln(5) - 2/ln(3)$

Explanation:

Separating the integrals, we get:

$\int_{0}^{1} 5^{x} d x - \int_{0}^{1} 3^{x} d x$ $int_0^1 5^xdx - int_0^1 3^xdx$

Now use the formula $\int (a^{x}) d x = \frac{a^{x}}{ln (a)}$ $int(a^x)dx = a^x/ln(a)$ .

${[\frac{5^{x}}{ln 5}]}_{0}^{1} - {[\frac{3^{x}}{ln 3}]}_{0}^{1}$ $[5^x/ln5]_0^1 - [3^x/ln3]_0^1$

$\frac{5}{ln (5)} - \frac{5^{0}}{ln (5)} - (\frac{3}{ln 3} - \frac{3^{0}}{ln 3})$ $5/ln(5) - 5^0/ln(5) - (3/ln3 - 3^0/ln3)$

$\frac{4}{ln (5)} - \frac{2}{ln (3)}$ $4/ln(5) - 2/ln(3)$

Hopefully this helps!

Science

Math

Humanities

... and beyond

How do you integrate $\int 5^{x} - 3^{x} d x$ $int 5^x-3^xdx$ from $[0, 1]$ $[0,1]$ ?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you integrate ∫5x−3xdxint 5^x-3^xdx from [0,1][0,1]?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

How do you integrate $\int 5^{x} - 3^{x} d x$ $int 5^x-3^xdx$ from $[0, 1]$ $[0,1]$ ?