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

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
∴ ∫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.
⇒ cos 3x = (frac{1}{3}) (frac{d}{dx}) (sin 3x)
⇒ cos 3x = (frac{d}{dx}) ((frac{1}{3}) sin 3x)
∴ 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.
⇒ e^2x = (frac{1}{2}) (frac{d}{dx}) (e^2x)
⇒ e^2x = (frac{d}{dx}) ((frac{1}{2}) e^2x)
∴ 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.
⇒ (ax + b)² = (frac{1}{3a}) (frac{d}{dx})(ax + b)³
⇒ (ax + b)² = (frac{d}{dx})[(frac{1}{3a})(ax + b)³]
∴ 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.
⇒ sin 2x = (frac{d}{dx})(-(frac{1}{2}) cos 2x)
and (frac{d}{dx})(4e^3x) = 4 × 3e^3x
⇒ 4e^3x = (frac{d}{dx})((frac{1}{3}) e^3x)
∴ 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. ∫(4e^3x + 1)dx
Solution:

Question 7.
∫x²(1 – (frac{1}{x^{2}}))dx
Solution:

Question 8.
∫(ax² + bx + c)dx
Solution:

Question 9.
∫(2x² + e^x) dx
Solution:

Question 10.
∫((sqrt{x}) – (frac{1}{sqrt{x}}))² dx
Solution:

Question 11.
∫(frac{x^{3}+5 x^{2}-4}{x^{2}})dx
Solution:

Question 12.
∫(frac{x^{3}+3 x+4}{sqrt{x}})dx
Solution:

Question 13.
∫(frac{x^{3}-x^{2}+x-1}{x-1})dx
Solution:

Question 14.
∫(1 – x)(sqrt{x}) dx
Solution:

Question 15.
∫(sqrt{x})(3x² + 2x + 3)dx
Solution:

Question 16.
∫(2x – 3cosx + e^x)dx
Solution:

Question 17.
∫(2x² – 3sinx + 5(sqrt{x}))dx
Solution:

Question 18.
∫secx(sec x + tan x)dx
Solution:

Question 19.
∫(frac{sec^{2}x}{cosec^{2}x})dx.
Solution:

Question 20.
∫(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})x² + 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:

⇒ Part(C) is the correct answer.

Question 22.
If (frac{d}{dx}) f(x) = 4x³ – (frac{3}{x^{4}}) such that f(2) = 0, then f(x) is
(A) x⁴ + (frac{1}{x^{3}}) – (frac{129}{8})
(B) x³ + (frac{1}{x^{4}}) + (frac{129}{8})
(C) x⁴ + (frac{1}{x^{3}}) + (frac{129}{8})
(D) x³ + (frac{1}{x^{4}}) – (frac{129}{8})
Solution:

Putting C = – (frac{129}{8}) in (1), we get
f(x) = x⁴ + (frac{1}{x^{3}}) – (frac{129}{8})
⇒ Part(A) is the correct answer.