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:

Question 2.

\(\int \frac{x^{3}}{\sqrt{1+x^{4}}} d x\)

Solution:

Question 3.

\(\int\left(e^{x}+e^{-x}\right)^{2}\left(e^{x}-e^{-x}\right) d x\)

Solution:

Question 4.

\(\int \frac{1+x}{x+e^{-x}} d x\)

Solution:

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

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:

Question 8.

\(\int \frac{2 x+6}{\sqrt{x^{2}+6 x+3}} d x\)

Solution:

Question 9.

\(\int \frac{1}{\sqrt{x}+x} d x\)

Solution:

Question 10.

\(\int \frac{1}{x\left(x^{6}+1\right)} d x\)

Solution: