The value of b > 3 for which 12∫₃ᵇ 1/(x² - 1)(x² - 4)dx = log ₑ( 49/40 ) , is equal to ___________.

Solution

Maths sir · 5 steps

Treat x² as one block, split into partial fractions, and both pieces become standard log integrals.

1. Partial fractions. Since (x²-1)-(x²-4)=3, the integrand splits as 1/3(1/(x²-4)-1/(x²-1)). Multiplying by 12 gives 4/(x²-4)-4/(x²-1).
2. Integrate each piece. Use ∫ dx/(x²-a²)=1/2a ln|(x-a)/(x+a)|. With a=2 and a=1, the antiderivative combines into one logarithm.

   F(x)=ln(x-2)/(x+2)-2 ln(x-1)/(x+1)=ln(x-2)(x+1)²/(x+2)(x-1)²

3. Apply the limits. At x=3 the fraction is (1 × 16)/(5 × 4)=4/5. So F(b)-ln 4/5=ln 49/40, which gives (b-2)(b+1)²/(b+2)(b-1)²=49/40×4/5=49/50.
4. Find b. The 49 in the target value suggests (b+1)²=49, so try b=6: (4 × 49)/(8 × 25)=196/200=49/50. It fits exactly, and 6>3 as required.
5. So. The required value is b= 6.

The idea to keep. When the integrand has only x² in it, treat x² as one variable for partial fractions. Then each piece is a standard 1/(x²-a²) log form.

Answer from the NTA final answer key. Solution by PrepWiser.