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

If this part uses an earlier result, revisit (a)(ii) Completing the square or open all working together.

(a)(iii)

Linear combination

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

Explore this idea

Hint

  1. Write x=−(2−x)+2x=-(2-x)+2.

Answer · (a)(iii)

−4x−x2−1+2arcsin⁡((x−2)/3)+C-\sqrt{4x-x^2-1}+2\arcsin((x-2)/\sqrt3)+C.

All a-integrands: 2−√3<x<2+√3.

Explore the idea

Open interactive exploration

Combining the two basic integrals

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
1.202
Below-axis contribution
0
Signed integral
1.202

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

Signed integral from 2.00 to x
1.202
antiderivative − F(2)
1.202
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 new integrand is a combination, so its antiderivative is the same combination of the two earlier answers.

Ideas to try
Compare with the previous two labs: what is the sum of them with the factors?