Find the definite integral ∫ (upper limit root sign 2 times a, lower limit 0) xdx / under root sign (3a2-x2)

Find the definite integral ∫ (upper limit root sign 2 times a, lower limit 0) xdx / under root sign (3a2-x2)

A:
Several of the questions you asked are of this type,
After removing the root sign with the substitution method, you can try, you can do it
Let x = √ 3 (a * Sint)

Find the definite integral ∫ (upper 2, lower 1) root sign (x ^ 2-1) / xdx

The indefinite integral of ∫ √ (x ^ 2-1) / xdx
Let √ (x ^ 2-1) = t, and the upper and lower limits are greater than 0
So x = √ (T ^ 2 + 1), DX = t / √ (T ^ 2 + 1) DT
So ∫ √ (x ^ 2-1) / xdx = ∫ T / √ (T ^ 2 + 1) * [T / √ (T ^ 2 + 1)] DT
=∫t^2/(t^2+1)dt=∫dt-∫1/(t^2+1)dt
=T-arctant + C can get t = √ (x ^ 2-1)
∫√(x^2-1)/xdx=√(x^2-1)-arctan√(x^2-1)+C
Then the upper and lower bounds of the integral can be obtained by subtracting them
∫ (up 2 down 1) √ (x ^ 2-1) / xdx = √ 3 - π / 3

What's the relationship between arctan and tan? RT

Arctan means arc tangent, for example: Tan 45 degree = 1, then
Arttan1 = 45 degrees is the operation of finding the inverse, just as the inverse operation of multiplication is division
Similarly, arcsin is arcsin
If sin30 degree = 1 / 2, arcsin1 / 2 = 30 degree
In addition, there are arccos and arccot, etc
That's basically it

Tan (arccos radical 2 / 2-pai / 6) = 3Q

Arccos radical 2 / 2 = 45 = Pai / 4
Tan (arccos radical 2 / 2-pai / 6) = Tan (PAI / 12)
If we want to change it, we'll use the auxiliary angle formula. Generally, it's OK

Cos α = - 2 root sign 2 / 3 and 90 degree < α < 270 degree, SEC α - Tan α = 3Q

cosa=-2√2/3
∴sina=±1/3
When a is in the second image limit
secα-tanα=1/cosa-sina/cosa=(1-sina)/cosa=(1-1/3)/-2√2/3=-√2/2
When a is in the third quadrant
secα-tanα=1/cosa-sina/cosa=(1-sina)/cosa=(1+1/3)/-2√2/3=-√2

How much is 2 / 3 root 9x + 6 times root sign 4 / 4 x

2 / 3 root number 9x + 6 times root number 4 / X
=2/3*3√x+6/2√x
=2√x+3√x
=5√x

(3 under the root sign - 6 under the root sign) * how much is 6 under the root sign + 2 / 3 under the root sign

(3 under root sign - 6 under root sign) * 6 under root sign + 2 / 3 under root sign
=3 * 6 - 6 + 2 / 3
=2-6 + 1 / 3 under 6

What's six fifths under the radical

Six fifths below the radical
=5 √ 30

12 plus three times the root 3 equals?

This is the simplest
If the approximate value is obtained
Then root 3 is about 1.732
So the original formula is about 17.196

What is root 12 plus root 3 plus root 27

√12+√3+√27
=2√3+√3+3√3
=6√3