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

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: