Where Mindful Learning Begins
Where Mindful Learning Begins
Chapter 7 Ex 7.7 covers the three special integral forms \sqrt{x^2\pm a^2} and \sqrt{a^2-x^2} — all 11 questions solved step by step, completing the square under the root in every single one.
This is a short but focused exercise: every question reduces to one of just three standard formulae once you complete the square under the root. When the quadratic under the root doesn't already look like a² − x², x² + a², or x² − a², rewrite it as (x + k)² plus or minus a constant first — the sign of that constant tells you which of the three formulae applies. The two MCQs at the end are quick checks on whether you can identify the right form at a glance under time pressure. Because there are only 11 questions here, it's worth drilling this exercise until the completing-the-square step becomes automatic — it resurfaces constantly in Ex 7.4 and again in definite integral problems later in the chapter.
Already in the form \sqrt{a^2-x^2} with a=2. Apply \displaystyle\int \sqrt{a^2-x^2}\,dx = \dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\sin^{-1}\left(\dfrac{x}{a}\right) directly.
Write as \sqrt{1-(2x)^2} and put t=2x, dx=\dfrac{dt}{2}.
The integral becomes \dfrac12\displaystyle\int \sqrt{1-t^2}\,dt = \dfrac12\left[\dfrac{t}{2}\sqrt{1-t^2}+\dfrac12\sin^{-1}t\right].
Complete the square: x^2+4x+6 = (x+2)^2+2.
Put t=x+2 and apply \displaystyle\int \sqrt{t^2+a^2}\,dt = \dfrac{t}{2}\sqrt{t^2+a^2}+\dfrac{a^2}{2}\log\left|t+\sqrt{t^2+a^2}\right| with a^2=2.
Complete the square: x^2+4x+1 = (x+2)^2-3.
Put t=x+2 and apply \displaystyle\int \sqrt{t^2-a^2}\,dt = \dfrac{t}{2}\sqrt{t^2-a^2}-\dfrac{a^2}{2}\log\left|t+\sqrt{t^2-a^2}\right| with a^2=3.
Complete the square: 1-4x-x^2 = 5-(x+2)^2. Put t=x+2 and apply the \sqrt{a^2-t^2} formula with a^2=5.
Complete the square: x^2+4x-5 = (x+2)^2-9. Put t=x+2 and apply the \sqrt{t^2-a^2} formula with a^2=9.
Complete the square: 1+3x-x^2 = \dfrac{13}{4}-\left(x-\dfrac32\right)^2. Put t=x-\dfrac32 and apply the \sqrt{a^2-t^2} formula with a^2=\dfrac{13}{4}.
Complete the square: x^2+3x = \left(x+\dfrac32\right)^2-\dfrac94. Put t=x+\dfrac32 and apply the \sqrt{t^2-a^2} formula with a^2=\dfrac94.
Rewrite \sqrt{1+\dfrac{x^2}{9}} = \dfrac13\sqrt{x^2+9}. Apply the \sqrt{x^2+a^2} formula with a=3, then scale by \dfrac13.
Already in standard form \sqrt{x^2+a^2} with a=1 — apply the formula directly.
Complete the square: x^2-8x+7 = (x-4)^2-9. Put t=x-4 and apply the \sqrt{t^2-a^2} formula with a^2=9.
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