Reuse two basic integrals and a radical substitution

(a)(i) Find ∫2−x4x−x2−1dx\int\frac{2-x}{\sqrt{4x-x^2-1}}dx. marks

(ii) Find ∫14x−x2−1dx\int\frac1{\sqrt{4x-x^2-1}}dx. marks

(iii) Using (i) and (ii), find ∫x4x−x2−1dx\int\frac x{\sqrt{4x-x^2-1}}dx. marks

(b)(i) By using u=xu=\sqrt x, find ∫x1+xdx\int\frac{\sqrt x}{1+\sqrt x}dx. marks

(ii) Find the exact value of ∫09ln⁡(1+x)dx\int_0^9\ln(1+\sqrt x)dx, leaving the answer aln⁡b−c/ba\ln b-c/b, where a,b,ca,b,c are integers to be determined. marks

[2016/VJC/Promo/11]

Total marks: 14

Solution

(a)(i)

Reverse chain rule

Find ∫2−x4x−x2−1 dx\int\dfrac{2-x}{\sqrt{4x-x^2-1}}\,dx.

Explore this idea

Hint

  1. The derivative of 4x−x2−14x-x^2-1 is 4−2x=2(2−x)4-2x=2(2-x).

Answer · (a)(i)

4x−x2−1+C\sqrt{4x-x^2-1}+C.

Explore the idea

Open interactive exploration

A radical with domain 2−√3 < x < 2+√3

Drag x. The shading shows the selected interval; the lower curves compare the numerical signed integral with the change in the antiderivative.

Upper limit xx = 2.83
Reset the controls

How the shaded areas add up

The limits run left to right. Area above the axis adds; area below the axis subtracts.

Above-axis contribution
0
Below-axis contribution
-0.209
Signed integral
-0.209

All three bars share one scale. Positive contributions extend right of zero; negative contributions extend left.

Signed integral from 2.00 to x
-0.209
√(4x−x²−1) − F(2)
-0.209
agrees at displayed precision
Formula − integral
0
0.00% of the integral’s magnitude
Compare the running integral on a graph
  • Integrand
  • Numerical signed integral from the start to x
  • Part of the integrand above the axis
  • Part of the integrand below the axis
  • F(x) − F(start), from the antiderivative
Key idea

The antiderivative is the radical itself, symmetric about x=2x=2.

Ideas to try
The integrand is an odd function about x=2x=2. What does that say about the area?