Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 7 Integrals Ex 7.1 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Maths Chapter 7 Integrals Ex 7.1

Find an antiderivative (or integral) of the following by the method of inspection:

Question 1.

sin 2x

Solution:

We know that \(\frac{d}{dx}\) cos 2x = – 2sin 2x.

or \(\frac{d}{dx}\) (- \(\frac{1}{2}\) cos 2x) = sin 2x

âˆ´ âˆ«sin 2x dx = – \(\frac{1}{2}\) cos 2x + C.

Question 2.

cos 3x

Solution:

2. We know that \(\frac{d}{dx}\) (sin 3x) = 3 cos 3x.

â‡’ cos 3x = \(\frac{1}{3}\) \(\frac{d}{dx}\) (sin 3x)

â‡’ cos 3x = \(\frac{d}{dx}\) (\(\frac{1}{3}\) sin 3x)

âˆ´ An antiderivative of cos 3x is \(\frac{1}{3}\) sin 3x + C.

By using free Taylor Series Calculator, you can easily find the approximate value of the integration function

Question 3.

e^{2x}

Solution:

We know that

\(\frac{d}{dx}\)(e^{2x}) = 2e^{2x}.

â‡’ e^{2x} = \(\frac{1}{2}\) \(\frac{d}{dx}\) (e^{2x})

â‡’ e^{2x} = \(\frac{d}{dx}\) (\(\frac{1}{2}\) e^{2x})

âˆ´ An antiderivative of e^{2x} is \(\frac{1}{2}\) e^{2x} + C.

Question 4.

(ax + b)^{2
}Solution:

We know that \(\frac{d}{dx}\) (ax + b)^{3} = 3a(ax + b)^{2}.

â‡’ (ax + b)^{2} = \(\frac{1}{3a}\) \(\frac{d}{dx}\)(ax + b)^{3}

â‡’ (ax + b)^{2} = \(\frac{d}{dx}\)[\(\frac{1}{3a}\)(ax + b)^{3}]

âˆ´ An antiderivative of (ax + b)^{2} = \(\frac{1}{3a}\)(ax + b)^{3} + C.

Question 5.

sin 2x – 4e^{3x
}Solution:

We know that \(\frac{d}{dx}\)(cos 2x) = – 2 sin 2x.

â‡’ sin 2x = \(\frac{d}{dx}\)(-\(\frac{1}{2}\) cos 2x)

and \(\frac{d}{dx}\)(4e^{3x}) = 4 Ã— 3e^{3x}

â‡’ 4e^{3x} = \(\frac{d}{dx}\)(\(\frac{1}{3}\) e^{3x})

âˆ´ An antiderivative of sin 2x – 4e^{3x} is – \(\frac{1}{2}\) cos 2x – \(\frac{4}{3}\)e^{3x} + C.

Find the following integrals:

Question

6. âˆ«(4e^{3x} + 1)dx

Solution:

Question 7.

âˆ«x^{2}(1 – \(\frac{1}{x^{2}}\))dx

Solution:

Question 8.

âˆ«(ax^{2} + bx + c)dx

Solution:

Question 9.

âˆ«(2x^{2} + e^{x}) dx

Solution:

Question 10.

âˆ«(\(\sqrt{x}\) – \(\frac{1}{\sqrt{x}}\))^{2} dx

Solution:

Question 11.

âˆ«\(\frac{x^{3}+5 x^{2}-4}{x^{2}}\)dx

Solution:

Question 12.

âˆ«\(\frac{x^{3}+3 x+4}{\sqrt{x}}\)dx

Solution:

Question 13.

âˆ«\(\frac{x^{3}-x^{2}+x-1}{x-1}\)dx

Solution:

Question 14.

âˆ«(1 – x)\(\sqrt{x}\) dx

Solution:

Question 15.

âˆ«\(\sqrt{x}\)(3x^{2} + 2x + 3)dx

Solution:

Question 16.

âˆ«(2x – 3cosx + e^{x})dx

Solution:

Question 17.

âˆ«(2x^{2} – 3sinx + 5\(\sqrt{x}\))dx

Solution:

Question 18.

âˆ«secx(sec x + tan x)dx

Solution:

Question 19.

âˆ«\(\frac{sec^{2}x}{cosec^{2}x}\)dx.

Solution:

Question 20.

âˆ«\(\frac{2-3 \sin x}{\cos ^{2} x}\) dx.

Solution:

Choose the correct answers in the following questions 21 and 22:

Question 21.

The antiderivative of (\(\sqrt{x}\) + \(\frac{1}{\sqrt{x}}\)) equals

(A) \(\frac{1}{3}\)x^{1/3} + 2x^{1/2} + C

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

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

(D) \(\frac{3}{2}\)x^{3/2} + \(\frac{1}{2}\)x^{1/2} + C

Solution:

â‡’ Part(C) is the correct answer.

Question 22.

If \(\frac{d}{dx}\) f(x) = 4x^{3} – \(\frac{3}{x^{4}}\) such that f(2) = 0, then f(x) is

(A) x^{4} + \(\frac{1}{x^{3}}\) – \(\frac{129}{8}\)

(B) x^{3} + \(\frac{1}{x^{4}}\) + \(\frac{129}{8}\)

(C) x^{4} + \(\frac{1}{x^{3}}\) + \(\frac{129}{8}\)

(D) x^{3} + \(\frac{1}{x^{4}}\) – \(\frac{129}{8}\)

Solution:

Putting C = – \(\frac{129}{8}\) in (1), we get

f(x) = x^{4} + \(\frac{1}{x^{3}}\) – \(\frac{129}{8}\)

â‡’ Part(A) is the correct answer.