PSEB 12th Class Maths Solutions Chapter 7 Integrals Ex 7.7

Punjab State Board PSEB 12th Class Maths Book Solutions Chapter 7 Integrals Ex 7.7 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7

Direction (1 – 9): Integrate the function.

Question 1.
\(\sqrt{4-x^{2}}\)
Solution.

Question 2.
\(\sqrt{1-4 x^{2}}\)
Solution.
Let I = ∫ \(\sqrt{1-4 x^{2}}\) dx

= ∫ \(\sqrt{(1)^{2}-(2 x)^{2}}\) dx

Let 2x = t
⇒ 2 dx = dt

Question 3.
\(\sqrt{x^{2}+4 x+6}\)
Solution.

Question 4.
\(\sqrt{x^{2}+4 x+1}\)
Solution.

Question 5.
\(\sqrt{1-4 x-x^{2}}\)
Solution.

Question 6.
\(\sqrt{x^{2}+4 x-5}\)
Solution.

Question 7.
\(\sqrt{1+3 x-x^{2}}\)
Solution.

Question 8.
\(\sqrt{x^{2}+3 x}\)
Solution.

Question 9.
\(\sqrt{1+\frac{x^{2}}{9}}\)
Solution.

Question 10.
∫ \(\sqrt{1+x^{2}}\) dx is equal to
(A) \(\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log \left|x+\sqrt{1+x^{2}}\right|\) + C

(B) \(\frac{2}{3}\left(1+x^{2}\right)^{\frac{3}{2}}\) + C

(C) \(\frac{2}{3} x\left(1+x^{2}\right)^{\frac{3}{2}}\) + C

(D) \(\frac{x^{2}}{2} \sqrt{1+x^{2}}+\frac{1}{2} x^{2} \log \left|x+\sqrt{1+x^{2}}\right|\) + C
Solution.
We know that, ∫ \(\sqrt{a^{2}+x^{2}}\) dx = \(\frac{x}{2} \sqrt{a^{2}+x^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|\) + C

Here, a² = 1

∴ ∫ \(\sqrt{1+x^{2}}\) dx = \(\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log \left|x+\sqrt{1+x^{2}}\right|\) + C

Question 11.
∫ \(\sqrt{x^{2}-8 x+7}\) is equal to
(A) \(\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}+9 \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|\) + C

(B) \(\frac{1}{2}(x+4) \sqrt{x^{2}-8 x+7}+9 \log \left|x+4+\sqrt{x^{2}-8 x+7}\right|\) + C

(C) \(\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}-3 \sqrt{2} \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|\) + C

(D) \(\frac{1}{2}(x-4) \sqrt{x^{2}-8 x+7}+\frac{9}{2} \log \left|x-4+\sqrt{x^{2}-8 x+7}\right|\) + C
Solution.

Hence, the correct answer is (D).