Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 5 Integration Ex 5.2 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2

Evaluate the following.

Question 1.
\(\int x \sqrt{1+x^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q1

Question 2.
\(\int \frac{x^{3}}{\sqrt{1+x^{4}}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q2

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2

Question 3.
\(\int\left(e^{x}+e^{-x}\right)^{2}\left(e^{x}-e^{-x}\right) d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q3

Question 4.
\(\int \frac{1+x}{x+e^{-x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q4

Question 5.
∫(x + 1)(x + 2)7(x + 3) dx
Solution:
Let I = ∫(x + 1)(x + 2)7(x + 3) dx
= ∫(x + 2)7 (x + 1)(x + 3) dx
= ∫(x + 2)7 [(x + 2) – 1][(x + 2) + 1] dx
= ∫(x + 2)7 [(x + 2)2 – 1] dx
= ∫[(x + 2)9 – (x + 2 )7] dx
= ∫(x + 2 )9 dx – ∫(x + 2)7 dx
= \(\frac{(x+2)^{10}}{10}\) – \(\frac{(x+2)^{8}}{8}\) + c

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2

Question 6.
\(\int \frac{1}{x \log x} d x\)
Solution:
Put log x = t
∴ \(\frac{1}{x}\) dx = dt
∴ \(\int \frac{d x}{x \cdot \log x}=\int \frac{1}{\log x} \cdot \frac{1}{x} d x\)
= ∫\(\frac{1}{t}\) dt
= log |t| + c
= log|log x| + c.

Question 7.
\(\int \frac{x^{5}}{x^{2}+1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q7

Question 8.
\(\int \frac{2 x+6}{\sqrt{x^{2}+6 x+3}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q8

Question 9.
\(\int \frac{1}{\sqrt{x}+x} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q9
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q9.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2

Question 10.
\(\int \frac{1}{x\left(x^{6}+1\right)} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Ex 5.2 Q10