Exercise 7.8 Class 12 Maths Integrals

NCERT Solutions Class 12 Maths Chapter 7 Ex 7.7 | LearnCBSEHub

💡 3 Key Formulas

Integrals of the form $\sqrt{Q(x)}$ where $Q(x)$ is quadratic.

$\int \sqrt{x^2 – a^2} dx = \frac{x}{2}\sqrt{x^2-a^2} – \frac{a^2}{2}\log|x+\sqrt{x^2-a^2}| + C$
$\int \sqrt{x^2 + a^2} dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log|x+\sqrt{x^2+a^2}| + C$
$\int \sqrt{a^2 – x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$

Questions 01 — 03

Basic Forms.

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

Format: $\sqrt{a^2 – x^2}$ with $a=2$.
$= \frac{x}{2}\sqrt{4-x^2} + \frac{4}{2}\sin^{-1}\frac{x}{2} + C$.
Ans: $\frac{x}{2}\sqrt{4-x^2} + 2\sin^{-1}\frac{x}{2} + C$.

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

$= \int \sqrt{1-(2x)^2} dx$. Let $2x=t \implies dx=dt/2$.
$\frac{1}{2} \int \sqrt{1-t^2} dt = \frac{1}{2} [\frac{t}{2}\sqrt{1-t^2} + \frac{1}{2}\sin^{-1}t]$.
Ans: $\frac{x}{2}\sqrt{1-4x^2} + \frac{1}{4}\sin^{-1}(2x) + C$.

3. $\int \sqrt{x^2+4x+6} dx$

Complete Square: $x^2+4x+6 = (x^2+4x+4) + 2 = (x+2)^2 + (\sqrt{2})^2$.
$\int \sqrt{(x+2)^2 + (\sqrt{2})^2} dx$. Use formula $\sqrt{X^2+a^2}$.
Ans: $\frac{x+2}{2}\sqrt{x^2+4x+6} + \log|x+2+\sqrt{x^2+4x+6}| + C$.

Questions 04 — 06

Completing the Square.

4. $\int \sqrt{x^2+4x+1} dx$

$x^2+4x+1 = (x+2)^2 – 3 = (x+2)^2 – (\sqrt{3})^2$.
Use formula $\sqrt{x^2-a^2}$ where $X=x+2, a=\sqrt{3}$.
Ans: $\frac{x+2}{2}\sqrt{x^2+4x+1} – \frac{3}{2}\log|x+2+\sqrt{x^2+4x+1}| + C$.

5. $\int \sqrt{1-4x-x^2} dx$

$1-(x^2+4x) = 1-(x^2+4x+4-4) = 5-(x+2)^2$.
Use formula $\sqrt{a^2-x^2}$ where $a=\sqrt{5}, X=x+2$.
Ans: $\frac{x+2}{2}\sqrt{1-4x-x^2} + \frac{5}{2}\sin^{-1}(\frac{x+2}{\sqrt{5}}) + C$.

6. $\int \sqrt{x^2+4x-5} dx$

$x^2+4x-5 = (x+2)^2 – 9 = (x+2)^2 – 3^2$.
Use formula $\sqrt{x^2-a^2}$.
Ans: $\frac{x+2}{2}\sqrt{x^2+4x-5} – \frac{9}{2}\log|x+2+\sqrt{x^2+4x-5}| + C$.

Questions 07 — 09

Advanced Completing the Square.

7. $\int \sqrt{1+3x-x^2} dx$

$1-(x^2-3x) = 1-(x^2-3x+\frac{9}{4}-\frac{9}{4}) = \frac{13}{4} – (x-\frac{3}{2})^2$.
$a = \frac{\sqrt{13}}{2}, X = x-\frac{3}{2}$. Formula $\sqrt{a^2-X^2}$.
$= \frac{x-3/2}{2}\sqrt{1+3x-x^2} + \frac{13/4}{2}\sin^{-1}\frac{x-3/2}{\sqrt{13}/2} + C$.
Ans: $\frac{2x-3}{4}\sqrt{1+3x-x^2} + \frac{13}{8}\sin^{-1}(\frac{2x-3}{\sqrt{13}}) + C$.

8. $\int \sqrt{x^2+3x} dx$

$x^2+3x = (x+\frac{3}{2})^2 – \frac{9}{4}$.
Formula $\sqrt{X^2-a^2}$.
$= \frac{x+3/2}{2}\sqrt{x^2+3x} – \frac{9/4}{2}\log|x+\frac{3}{2}+\sqrt{x^2+3x}| + C$.
Ans: $\frac{2x+3}{4}\sqrt{x^2+3x} – \frac{9}{8}\log|x+\frac{3}{2}+\sqrt{x^2+3x}| + C$.

9. $\int \sqrt{1+\frac{x^2}{9}} dx$

$= \frac{1}{3} \int \sqrt{9+x^2} dx$. Formula $\sqrt{a^2+x^2}$ with $a=3$.
$= \frac{1}{3} [ \frac{x}{2}\sqrt{9+x^2} + \frac{9}{2}\log|x+\sqrt{9+x^2}| ] + C$.
Ans: $\frac{x}{6}\sqrt{9+x^2} + \frac{3}{2}\log|x+\sqrt{9+x^2}| + C$.

Questions 10 — 11

Multiple Choice Questions.

10. $\int \sqrt{1+x^2} dx$

Direct formula $\int \sqrt{a^2+x^2} dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log|x+\sqrt{x^2+a^2}|$.
Here $a=1$.
Result: $\frac{x}{2}\sqrt{1+x^2} + \frac{1}{2}\log|x+\sqrt{1+x^2}| + C$.

Correct Option: (A)

11. $\int \sqrt{x^2-8x+7} dx$

$x^2-8x+7 = (x-4)^2 – 16 + 7 = (x-4)^2 – 3^2$.
Formula $\sqrt{X^2-a^2}$.
$= \frac{x-4}{2}\sqrt{x^2-8x+7} – \frac{9}{2}\log|x-4+\sqrt{x^2-8x+7}| + C$.

Correct Option: (D)