Reuse two basic integrals and a radical substitution

\int\frac{2-x}{\sqrt{4x-x^2-1}}dx

Total marks: 14

Solution

(b)(i)

By using $u=\sqrt x$ , find $\int\dfrac{\sqrt x}{1+\sqrt x}\,dx$ .

Hint

$x=u^2$ so Differentiate $x=u^2$ to obtain $dx=2u\,du$ . Equivalently, for $x>0$ , differentiate $u=\sqrt x$ and invert its derivative. Use $u\ge0$ ; then $1+u>0$ , which justifies the logarithm without modulus.
Divide to get a polynomial plus a simple fraction.

Answer · (b)(i)

x-2\sqrt x+2\ln(1+\sqrt x)+C

Explore the idea

Open interactive exploration

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 = 4.50

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

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

Signed integral from 0.00 to x

2.534

antiderivative − F(0)

2.534

agrees at displayed precision

Formula − integral

0.00% of the integral’s magnitude

Compare the running integral on a graph

Key idea

For large $x$ the area grows almost linearly, because the integrand tends to 1.

Ideas to try

The integrand approaches 1 for large

x

: what does that make the area?