# Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.1

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Indefinite Integration Ex 3.1 Questions and Answers.

## Maharashtra State Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.1

I. Integrate the following functions w.r.t. x:

(i) x3 + x2 – x + 1
Solution:

(ii) $$x^{2}\left(1-\frac{2}{x}\right)^{2}$$
Solution:

(iii) $$3 \sec ^{2} x-\frac{4}{x}+\frac{1}{x \sqrt{x}}-7$$
Solution:

(iv) $$2 x^{3}-5 x+\frac{3}{x}+\frac{4}{x^{5}}$$
Solution:

(v) $$\frac{3 x^{3}-2 x+5}{x \sqrt{x}}$$
Solution:

II. Evaluate:

(i) ∫tan2 x . dx
Solution:

(ii) $$\int \frac{\sin 2 x}{\cos x} \cdot d x$$
Solution:

(iii) $$\int \frac{\sin x}{\cos ^{2} x} \cdot d x$$
Solution:

(iv) $$\int \frac{\cos 2 x}{\sin ^{2} x} \cdot d x$$
Solution:

(v) $$\int \frac{\cos 2 x}{\sin ^{2} x \cdot \cos ^{2} x} \cdot d x$$
Solution:

= -cot x – tan x + c

(vi) $$\int \frac{\sin x}{1+\sin x} \cdot d x$$
Solution:

(vii) $$\int \frac{\tan x}{\sec x+\tan x} \cdot d x$$
Solution:

(viii) $$\int \sqrt{1+\sin 2 x} \cdot d x$$
Solution:

(ix) $$\int \sqrt{1-\cos 2 x} \cdot d x$$
Solution:

(x) ∫sin 4x cos 3x dx
Solution:

III. Evaluate:

(i) $$\int \frac{x}{x+2} \cdot d x$$
Solution:

(ii) $$\int \frac{4 x+3}{2 x+1} \cdot d x$$
Solution:

(iii) $$\int \frac{5 x+2}{3 x-4} \cdot d x$$
Solution:

(iv) $$\int \frac{x-2}{\sqrt{x+5}} \cdot d x$$
Solution:

(v) $$\int \frac{2 x-7}{\sqrt{4 x-1}} \cdot d x$$
Solution:

(vi) $$\int \frac{\sin 4 x}{\cos 2 x} \cdot d x$$
Solution:

(vii) $$\int \sqrt{1+\sin 5 x} \cdot d x$$
Solution:

(viii) ∫cos2 x . dx
Solution:

(ix) $$\int \frac{2}{\sqrt{x}-\sqrt{x+3}} \cdot d x$$
Solution:

(x) $$\int \frac{3}{\sqrt{7 x-2}-\sqrt{7 x-5}} \cdot d x$$
Solution:

IV.

Question 1.
If f'(x) = x – $$\frac{3}{x^{3}}$$, f(1) = $$\frac{11}{2}$$, find f(x).
Solution:
By the definition of integral,