Exercise 7.1 Practice
Integration Basics & Method of Inspection – NCERT Solutions
Q1
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Find an anti-derivative (or integral) of the function $\sin 2x$ by the method of inspection.
We look for a function whose derivative is $\sin 2x$.
$\frac{d}{dx}(\cos 2x) = -2\sin 2x \Rightarrow \sin 2x = -\frac{1}{2}\frac{d}{dx}(\cos 2x)$.
Therefore, an anti-derivative is $-\frac{1}{2}\cos 2x$.
$\boxed{-\frac{1}{2}\cos 2x}$
Q2
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Find an anti-derivative (or integral) of the function $\cos 3x$ by the method of inspection.
We look for a function whose derivative is $\cos 3x$.
$\frac{d}{dx}(\sin 3x) = 3\cos 3x \Rightarrow \cos 3x = \frac{1}{3}\frac{d}{dx}(\sin 3x)$.
Therefore, an anti-derivative is $\frac{1}{3}\sin 3x$.
$\boxed{\frac{1}{3}\sin 3x}$
Q3
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Find an anti-derivative (or integral) of the function $e^{2x}$ by the method of inspection.
We know that $\frac{d}{dx}(e^{2x}) = 2e^{2x}$.
So, $e^{2x} = \frac{1}{2}\frac{d}{dx}(e^{2x})$.
Therefore, an anti-derivative is $\frac{1}{2}e^{2x}$.
$\boxed{\frac{1}{2}e^{2x}}$
Q4
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Find an anti-derivative (or integral) of the function $(ax + b)^2$ by the method of inspection.
We look for a function whose derivative is $(ax+b)^2$.
Consider $(ax+b)^3$. $\frac{d}{dx}(ax+b)^3 = 3(ax+b)^2 \cdot a = 3a(ax+b)^2$.
So, $(ax+b)^2 = \frac{1}{3a}\frac{d}{dx}(ax+b)^3$.
$\boxed{\frac{1}{3a}(ax+b)^3}$
Q5
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Find an anti-derivative (or integral) of the function $\sin 2x - 4e^{3x}$ by the method of inspection.
Anti-derivative of $\sin 2x$ is $-\frac{1}{2}\cos 2x$.
Anti-derivative of $e^{3x}$ is $\frac{1}{3}e^{3x}$.
Combining them: $-\frac{1}{2}\cos 2x - 4(\frac{1}{3}e^{3x})$.
$\boxed{-\frac{1}{2}\cos 2x - \frac{4}{3}e^{3x}}$
Q6
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Find the integral: $\int (4e^{3x} + 1) dx$.
$\int 4e^{3x} dx + \int 1 dx$.
$4 \cdot \frac{e^{3x}}{3} + x + C$.
$\boxed{\frac{4}{3}e^{3x} + x + C}$
Q7
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Find the integral: $\int x^2(1 - \frac{1}{x^2}) dx$.
Multiply $x^2$ inside: $\int (x^2 - 1) dx$.
Integrate term by term: $\frac{x^3}{3} - x + C$.
$\boxed{\frac{x^3}{3} - x + C}$
Q8
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Find the integral: $\int (ax^2 + bx + c) dx$.
Integrate each term: $a\int x^2 dx + b\int x dx + \int c dx$.
$a\frac{x^3}{3} + b\frac{x^2}{2} + cx + C$.
$\boxed{\frac{ax^3}{3} + \frac{bx^2}{2} + cx + C}$
Q9
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Find the integral: $\int (2x^2 + e^x) dx$.
Integrate term by term: $2\int x^2 dx + \int e^x dx$.
$2\frac{x^3}{3} + e^x + C$.
$\boxed{\frac{2}{3}x^3 + e^x + C}$
Q10
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Find the integral: $\int (\sqrt{x} - \frac{1}{\sqrt{x}})^2 dx$.
Expand: $(\sqrt{x})^2 - 2(\sqrt{x})(\frac{1}{\sqrt{x}}) + (\frac{1}{\sqrt{x}})^2 = x - 2 + \frac{1}{x}$.
Integrate: $\int x dx - \int 2 dx + \int \frac{1}{x} dx$.
$\boxed{\frac{x^2}{2} - 2x + \log|x| + C}$
Q11
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Find the integral: $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$.
Divide each term by $x^2$: $x + 5 - 4x^{-2}$.
Integrate: $\frac{x^2}{2} + 5x - 4\frac{x^{-1}}{-1}$.
$\boxed{\frac{x^2}{2} + 5x + \frac{4}{x} + C}$
Q12
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Find the integral: $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$.
Divide by $x^{1/2}$: $x^{5/2} + 3x^{1/2} + 4x^{-1/2}$.
Integrate: $\frac{x^{7/2}}{7/2} + 3\frac{x^{3/2}}{3/2} + 4\frac{x^{1/2}}{1/2}$.
$\boxed{\frac{2}{7}x^{7/2} + 2x^{3/2} + 8\sqrt{x} + C}$
Q13
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Find the integral: $\int \frac{x^3 - x^2 + x - 1}{x-1} dx$.
Factor numerator: $x^2(x-1) + 1(x-1) = (x^2+1)(x-1)$.
Simplify: $\int (x^2+1) dx$.
$\boxed{\frac{x^3}{3} + x + C}$
Q14
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Find the integral: $\int (1-x)\sqrt{x} dx$.
Expand: $x^{1/2} - x^{3/2}$.
Integrate: $\frac{x^{3/2}}{3/2} - \frac{x^{5/2}}{5/2}$.
$\boxed{\frac{2}{3}x^{3/2} - \frac{2}{5}x^{5/2} + C}$
Q15
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Find the integral: $\int \sqrt{x}(3x^2 + 2x + 3) dx$.
Expand: $3x^{5/2} + 2x^{3/2} + 3x^{1/2}$.
Integrate: $3\frac{x^{7/2}}{7/2} + 2\frac{x^{5/2}}{5/2} + 3\frac{x^{3/2}}{3/2}$.
$\boxed{\frac{6}{7}x^{7/2} + \frac{4}{5}x^{5/2} + 2x^{3/2} + C}$
Q16
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Find the integral: $\int (2x - 3\cos x + e^x) dx$.
Integrate term by term: $2\frac{x^2}{2} - 3\sin x + e^x$.
$\boxed{x^2 - 3\sin x + e^x + C}$
Q17
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Find the integral: $\int (2x^2 - 3\sin x + 5\sqrt{x}) dx$.
Integrate: $2\frac{x^3}{3} - 3(-\cos x) + 5\frac{x^{3/2}}{3/2}$.
$\boxed{\frac{2}{3}x^3 + 3\cos x + \frac{10}{3}x^{3/2} + C}$
Q18
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Find the integral: $\int \sec x (\sec x + \tan x) dx$.
Expand: $\sec^2 x + \sec x \tan x$.
Integrate: $\tan x + \sec x$.
$\boxed{\tan x + \sec x + C}$
Q19
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Find the integral: $\int \frac{\sec^2 x}{\csc^2 x} dx$.
Simplify: $\frac{1/\cos^2 x}{1/\sin^2 x} = \frac{\sin^2 x}{\cos^2 x} = \tan^2 x$.
Use identity: $\tan^2 x = \sec^2 x - 1$.
Integrate: $\int (\sec^2 x - 1) dx = \tan x - x$.
$\boxed{\tan x - x + C}$
Q20
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Find the integral: $\int \frac{2 - 3\sin x}{\cos^2 x} dx$.
Split: $\frac{2}{\cos^2 x} - \frac{3\sin x}{\cos^2 x} = 2\sec^2 x - 3\sec x \tan x$.
Integrate: $2\tan x - 3\sec x$.
$\boxed{2\tan x - 3\sec x + C}$
Q21
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The anti-derivative of $\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)$ 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) $\frac{3}{2}x^{3/2} + \frac{1}{2}x^{1/2} + C$
(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) $\frac{3}{2}x^{3/2} + \frac{1}{2}x^{1/2} + C$
Integrate $x^{1/2} + x^{-1/2}$.
$\frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} = \frac{2}{3}x^{3/2} + 2x^{1/2}$.
$\boxed{\text{(C) } \frac{2}{3}x^{3/2} + 2x^{1/2} + C}$
Q22
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If $\frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}$ such that $f(2) = 0$, then $f(x)$ is:
(A) $x^4 + x^{-3} - \frac{129}{8}$
(B) $x^3 + x^{-4} + \frac{129}{8}$
(C) $x^4 + x^{-3} + \frac{129}{8}$
(D) $x^3 + x^{-4} - \frac{129}{8}$
(A) $x^4 + x^{-3} - \frac{129}{8}$
(B) $x^3 + x^{-4} + \frac{129}{8}$
(C) $x^4 + x^{-3} + \frac{129}{8}$
(D) $x^3 + x^{-4} - \frac{129}{8}$
$f(x) = \int (4x^3 - 3x^{-4}) dx = x^4 + x^{-3} + C$.
$f(2) = 16 + \frac{1}{8} + C = 0 \Rightarrow C = -\frac{129}{8}$.
$\boxed{\text{(A) } x^4 + \frac{1}{x^3} - \frac{129}{8}}$