Establish a reduction formula for each of the following integrals:
1 sinⁿ x dx
2 tanⁿ 2x dx
3 (x+1ⁿ e2x dx
4 cosⁿ 2θ dθ
5 xⁿ eax dx
6 x (lnx)ⁿ dx
7 coshⁿx dx
8 secⁿ x dx
Use a reduction method to find each of the following indefinite integrals
9 cos ⁶ x dx
10 sin ⁷ x dx
11 (1+x)ⁿ e4x dx
12 cos⁴(ax+b) dx
13 sin⁴ (π/4 + 3θ) dθ
14 tan⁵ x dx
15 x³ sin x dx
16 sinh⁵ x dx
17 sec⁴ x dx
18 x⁶ e-x dx
19 x (lnx)³ dx
20 cosh⁴ 3x dx
21 Prove that, if In = ∫xⁿ(1+x³)⁷ dx then
In = 1/(n+22) [xn-2(1+x³)⁸ – (n-2)In-3]
Hint: use xⁿ = x²xn-2
Hence or otherwise determine ∫x⁵(1+x³)⁷ 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 ∫cos⁶x/sinx dx
Q9 onwards are likely to have long answers. Be prepared to find a value between limits.
Answers? Not here !!
