Challenge (1 Viewer)

[Forbidden.]

agreed.

inspection is pro

yeah much simpler than making a substitution

Solve these integrals by inspection WITHOUT using subsituttion.
4u maths kiddies wont need to use integration by parts because these integrals can be solved without it.
what a good time to solve this, before the hsc.



1. ∫ xe^2x2 dx

2. ∫ xsin(x²) dx

3. ∫ x²cos(2x³) dx

4. ∫ x/(5x² - 11) dx

5. ∫ sin x cos³x dx

6. ∫ 1/(x lnx) dx

7. ∫ (x + 2)/√(x² + 4x + 7) dx

8. ∫ x√(1 + x²) dx

9. ∫ x²√(9 - 4x³) dx

10. ∫ x² / √(9 - 4x³) dx

11. ∫ x³ / (1 + x⁴)³ dx

12. ∫ sec²x / tan³x dx

(Hint: (tan x)ⁿ = tanⁿx)

13. ∫ cos x / sin³x dx

14. ∫ e^2x(4 + 3e^2x)^1/3 dx

15. ∫ 1/(x(ln x)⁵) dx

 

[tommykins]

回复: Challenge

Doing first 5 to stimulate mind before doing past papers

1. e^(2x^2)/4
2. -cos(x^2)/2
3. sin(2x^3)/6
4. ln(5x^2-11)/10
5. (cosx)^4/4

 

[conics2008]

Re: 回复: Challenge

1) xe^2x^2 = 1/4 e^2x^2 from d/dx(2x^2)
2) xsin(x^2)= -1/2cos(x^2) from d/dx(x^2)
3) x^2cos(3x^2) = 1/6sin(3x^2) from d/dx(3x^2)
4) x/5x^2-11 = 1/10 ln(5x^2-11) from 10x --> 1/10
5) cos^3(x)sin(x) = -1/4cos^4(x) using f(x)f'(x)=f(x)^n+1/n+1
6) x/lnx by rearranging = 1/x/lnx = ln(lnx) use f'(x)/f(x)
7) x+2/root x^2+4x+7 = 1/2ln[root x^2+4x+7] using same as above
8) xroot 1+x^2 = 1/3(1+x^2)^3/2 looking at the fact that power must be 3/2 and dividing by 3 in turn to eliminate 3 and 2 and leaving x.
9) x^2(9-4x^3)^-1/2 look between relation of -4x^3 and 1/2 and x^2 gives us hmm... -1/6 that will cancel out the 12 soo it becomes >>>
-1/6(9-4x^3)^1/2
10) x^2/root (9-4x^3) >> write as x^2(9-4x^3)^-1/2 hence S = -1/6(9-4x^30^1/2. relations between 1/2 12 and 1
11) x^3(1+x^4)^-3 S= -1/8 (1+x^4)^-2 relation between 4 1 and -3
12) I cant seem to manipulate them.. except for tan^2(x)=sec^2-1.... need help here.
13) -1/2 (sinx)^-2 .. comes from the same one as 5
14) 1/8(4+3e^2x)^4/3 looking at 4/3 and 6 and making them equal 1
15) same as 6 which gives us 1/6ln(ln(x)^6)

Thanks Stranger.

 

[minijumbuk]

How can you do these without substitution =\

I'm so noob xD

 

[tommykins]

回复: Re: Challenge

How can you do these without substitution =\

I'm so noob xD

Look for patterns.

int f'x/fx = ln fx
int f'x.f^n x = f^n+1 x

etc. etc.

 

[conics2008]

Re: 回复: Re: Challenge

hey

can you post some more topics by topics question.

 

[Forbidden.]

I'll post up the answers sometimes kids.
You should've used spoiler tags
[SPOILER ][/SPOILER ]

remove the spaces.

How can you do these without substitution =\

I'm so noob xD

Well it looks like you don't know your derivatives and integrals of elementary functions (trigonometric, exponential, logarithmic, etc.) back-to-front.

Look for patterns.

int f'x/fx = ln fx
int f'x.f^n x = f^n+1 x

etc. etc.

HA!

edit: oops didnt realise this was 2 unit, good practice for ext 2 though

complex numbers
1.
prove that [|Re(z)| + |Im(z)|] / sqrt(2) < |z| for any complex number z

2.
a) given that z = cos(x) + i sin(x), show that z^n + z^(-n) = 2 cos(nx)
b) hence express (cos x)^6 in terms of nx
c) hence evaluate I (cos x)^6 dx (limits: 0 -> pi/6)

polynomials
1.
a polynomial P(x) is given by P(x) = x^5 - 5px + q, where p and q are real
a) by considering the turning points, prove that if p < 0, P(x) = 0 has only one real root
b) prove that P(x) = 0 has 3 distinct real roots if q^4 < 256p^5

2.
a and b are the roots of z^2 - 2z + 2 = 0. if cot @ = x + 1, prove that:
[(x + a)^n - (x + b)^n] / (a - b) = sin(n@) / (sin @)^n

mechanics
1.
a model plane of mass 5kg, attached to the end of a light inelastic wire, the other end of which is held fixed, flies in a horizontal circle at an elevation of 30 degrees. the upwards force of the air on this plane is twice the weight of the plane. find:
a) the angular velocity w of the plane about the circle
b) the tension in the wire

2.
a disk of radius 3m rotates at an angular velocity w = (1.6 + 1.2t) rad/s, where t is in seconds. at the instant t = 2, determine:
a) the angular acceleration
b) the speed, and the components of the acceleration of a point on the edge of the disk

3.
a small ring C can move freely on a light inextensible string. the two ends of the string are attached to points A and B, where A is vertically above B and at a distance c from it. when the ring C is describing a horizontal circle with constant angular velocity w, the distances of C from A and B are b and a respectively. show that:
2gc (a + b) = w^2 (a - b) [c^2 - (a + b)^2]

harder 3 unit
1.
a sequence {a_n} is given by a₁ = 2^(1/2), a_n+1 = (2 + a_n)^(1/2)
a) by induction show that {a_n} is increasing, but always less than 3
b) find the limit as n approaches infinity of a_n

2.
let a and b be positive numbers with a > b. let a₁ be their arithmetic mean and b₁ their geometric mean. use mathematical induction to show that:
a_n > a_n+1 > b_n+1 > b_n

Shit I still can't solve harder complex number questions given the calibre of Mathematics in uni is much greater than that of HSC.
Mechanics I'll just need some scrap paper kthnx.
thnx 4 shoving 4 unit in 2 unit thread.