Reduction Formulae | | DJS

Reduction Formulae

Establish a reduction formula for each of the following integrals:

1    sinx dx

2    tan2x dx

3    (x+1ⁿ e2x dx

4    cos2θ dθ

5    x eax dx

6    x (lnx) dx

7    coshⁿx dx

8    secⁿ x dx

Use a reduction method to find each of the following indefinite integrals

9    cos x dx

10   sin x dx

11   (1+x) e4x dx

12   cos(ax+b) dx

13   sin (π/4 + 3θ) dθ

14   tanx dx

15   x³ sin x dx

16   sinh x dx

17   sec x dx

18   x e-x dx

19   x (lnx)³ dx

20   cosh 3x dx

21   Prove that, if In = ∫x(1+x³) dx then

    In = 1/(n+22) [xn-2(1+x³)(n-2)In-3]

Hint: use x = x²xn-2
 Hence or otherwise determine  ∫x(1+x³) dx

22   If  In = ∫ cos2nx / sin x    dx then write down a similar expression for In+1.

By using the identity cos²x + sin²x = 1, prove (2n+1) In+1 = (2n+1) In + cos2n+1x

Hence or otherwise determine  ∫cosx/sinx dx

Q9 onwards are likely to have long answers. Be prepared to find a value between limits.

Answers? Not here !!

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