Find indefinite integral ∫ [(e ^ (3x) - 1) / (e ^ (x) - 1)] DX How to get ∫ [(e ^ (x) - 1) (e ^ (2x) + e ^ (x) + 1) / (e ^ (x) - 1)] DX

Find indefinite integral ∫ [(e ^ (3x) - 1) / (e ^ (x) - 1)] DX How to get ∫ [(e ^ (x) - 1) (e ^ (2x) + e ^ (x) + 1) / (e ^ (x) - 1)] DX

It is obtained from the cubic difference formula (a ^ 3-1) = (A-1) (a ^ 2 + A + 1)

Find indefinite integral ∫ (e ^ (3x) + 1) / (e ^ (x) + 1)

The key to this problem is to deform the integrand identity
Original formula = ∫ (e ^ (3x) - e ^ x + e ^ x + 1) / (e ^ (x) + 1)
=∫e^x(e^x-1)dx+∫1dx
=∫e^(2x)dx-∫e^xdx+x
=1/2∫e^(2x)d(2x)-e^x+x
=e^(2x)/2-e^x+x+C

√ (3x-2) / X indefinite integral

Let t = √ (3x-2), get x = (T ^ 2 + 2) / 2, DX = t DT
∫√(3x-2)/xdx
=∫2t/(t^2+2)·t dt
=2∫(t^2)/(t^2+2) dt
=2∫(t^2+2-2)/(t^2+2) dt
=2[∫1dt-2∫1/(t^2+2)]dt
=2∫1dt-4√2∫1/[(t/√2)^2+1]d(t/√2)
=2t-4√2 arctan(t/√2)+C
=2√(3x-2)-4√2arctan√(3x/2-1)+C

The problem of judging triangle shape by sine theorem and cosine theorem In the known triangle ABC, acosb = bcosa, then the triangle is () A isosceles triangle b right triangle C isosceles or right triangle D obtuse triangle I did this a / b = cosa / CoSb = Sina / SINB So Sina / SINB = cosa / CoSb, and then a = b = 45 degrees, so it's an isosceles triangle or a right triangle, but the answer is a isosceles If both a and B are equal to 45 degrees, a + B is equal to 90 degrees, then C is 90 degrees. Isn't it a right triangle? Why not choose C?

sinA/sinB=cosA/cosB
sinAcosB-cosAsinB=0
Get sin (a-b) = 0
- 180 < - B < 0 because of 00
So - 180 gets A-B = 0 from sin (a-b) = 0
A=B
So we can't draw a right angle conclusion

In △ ABC, if Sina: SINB: sinc = 2:3:4, then COSC is equal to () A. 2 three B. −2 three C. −1 three D. −1 four

From sine theorem; sinA：sinB：sinC=a：b：c=2：3：4
A = 2K, B = 3k, C = 4K (k > 0) can be set
From the cosine theorem, COSC = A2 + B2 − C2
2ab=4k2+9k2−16k2
2•2k•3k＝−1
four
Therefore: D

Know how to find the angle of the three sides of a triangle according to the sine theorem or cosine theorem If you know cosa = 0.258819045 How to find angle a

Three methods: 1. Look up the table
2. Press Shift + cos + 0.258819045 on the calculator
3. Check the following scientific type on the calculator of the computer system - tick in front of inv - enter data - press cos