actever6a

2021-11-10

Evaluate the integrals in $\int \frac{\mathrm{ln}\left(x\right)dx}{x\sqrt{{\mathrm{ln}}^{2}\left(x\right)+1}}$

Pulad1971

Beginner2021-11-11Added 22 answers

Step 1

Consider the following integral:

$\int \frac{\mathrm{ln}\left(x\right)}{x\sqrt{{\mathrm{ln}}^{2}\left(x\right)+1}}dx=I$

Substitute${\mathrm{ln}}^{2}\left(x\right)+1=u\to \frac{2\mathrm{ln}\left(x\right)}{x}dx=du$ in the above integral:

$I=\int \frac{1}{2\sqrt{u}}du$

$=\frac{1}{2}\int {\left(u\right)}^{-\frac{1}{2}}du$

$=\frac{1}{2}\times \frac{{u}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+C$

$=\frac{1}{2}\times \frac{{u}^{\frac{1}{2}}}{\frac{1}{2}}+C$

$=\sqrt{u}+C$

Step 2

Substitute$u={\mathrm{ln}}^{2}\left(x\right)+1$ in the above equation:

$I=\sqrt{{\mathrm{ln}}^{2}\left(x\right)+1}+C$

Hence, the solution is$\sqrt{{\mathrm{ln}}^{2}\left(x\right)+1}+C$ .

Consider the following integral:

Substitute

Step 2

Substitute

Hence, the solution is

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1)?

How to expand and simplify $2(3x+4)-3(4x-5)$?

Find an equation equivalent to ${x}^{2}-{y}^{2}=4$ in polar coordinates.

How to graph $r=5\mathrm{sin}\theta$?

How to find the length of a curve in calculus?

When two straight lines are parallel their slopes are equal.

A)True;

B)FalseIntegration of 1/sinx-sin2x dx

Converting percentage into a decimal. $8.5\%$

Arrange the following in the correct order of increasing density.

Air

Oil

Water

BrickWhat is the exact length of the spiraling polar curve $r=5{e}^{2\theta}$ from 0 to $2\pi$?

What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?

What is the slope of the tangent line of $r=-2\mathrm{sin}\left(3\theta \right)-12\mathrm{cos}\left(\frac{\theta}{2}\right)$ at $\theta =\frac{-\pi}{3}$?

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?

Use the summation formulas to rewrite the expression $\Sigma \frac{2i+1}{{n}^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000.

How to calculate the right hand and left hand riemann sum using 4 sub intervals of f(x)= 3x on the interval [1,5]?