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Q2:
Find an anti derivative (or integral) of the following functions by the method of inspection.
2. $\cos 3x$
Solution :
Step 1: Identifying the Derivative Relationship
To find the anti-derivative (integral) of the function $f(x) = \cos 3x$ by the method of inspection, we must identify a function $F(x)$ such that its derivative yields the original function:
$\frac{d}{dx} [F(x)] = \cos 3x$ [Definition of Anti-derivative]
Step 2: Selecting the Trial Function
We know from the standard trigonometric derivatives that the derivative of the sine function results in a cosine function:
$\frac{d}{dx} (\sin ax) = a \cos ax$ [Chain Rule of Differentiation]
In our case, $a = 3$. Therefore, let us consider the function $\sin 3x$.
Step 3: Calculating the Derivative of the Trial Function
Differentiating $\sin 3x$ with respect to $x$:
$\frac{d}{dx} (\sin 3x) = \cos 3x \cdot \frac{d}{dx} (3x)$ [Applying the Chain Rule]
$\frac{d}{dx} (\sin 3x) = \cos 3x \cdot 3$
$\frac{d}{dx} (\sin 3x) = 3 \cos 3x$
Step 4: Isolating the Target Function
To isolate $\cos 3x$ as requested by the inspection method, divide both sides of the equation by $3$:
$\frac{1}{3} \cdot \frac{d}{dx} (\sin 3x) = \cos 3x$
Using the linearity property of derivatives, $\frac{d}{dx} [k \cdot f(x)] = k \cdot \frac{d}{dx} [f(x)]$, we can move the constant inside the derivative operator:
$\frac{d}{dx} \left( \frac{1}{3} \sin 3x \right) = \cos 3x$
Step 5: Visualizing the Relationship
Final Solution: The anti-derivative of $\cos 3x$ is $\frac{1}{3} \sin 3x$.
More Questions from Class 12 Mathematics Integrals EXERCISE 7.1
- Q1: Find an anti derivative (or integral) of the following functions by the method of inspection. 1. $\sin 2x$
- Q3: Find an anti derivative (or integral) of the following functions by the method of inspection. 3. $e^{2x}$
- Q4: Find an anti derivative (or integral) of the following functions by the method of inspection. 4. $(ax + b)^2$
- Q5: Find an anti derivative (or integral) of the following functions by the method of inspection. 5. $\sin 2x - 4e^{3x}$
- Q6: Find the following integrals in Exercises 6 to 20: 6. $\int (4e^{3x} + 1) dx$
- Q7: Find the following integrals in Exercises 6 to 20: 7. $\int x^2(1 - \frac{1}{x^2}) dx$
- Q8: Find the following integrals 8. $\int (ax^2 + bx + c) dx$
- Q9: Find the following integrals 9. $\int (2x^2 + e^x) dx$
- Q10: Find the following integrals 10. $\int (\sqrt{x} - \frac{1}{\sqrt{x}})^2 dx$
- Q11: Find the following integrals 11. $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$
- Q12: Find the following integrals 12. $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$
- Q13: Find the following integrals 13. $\int \frac{x^3 - x^2 + x - 1}{x - 1} dx$
- Q14: Find the following integrals 14. $\int (1 - x)\sqrt{x} dx$
- Q15: Find the following integrals 15. $\int \sqrt{x} (3x^2 + 2x + 3) dx$
- Q16: Find the following integrals 16. $\int (2x - 3\cos x + e^x) dx$
- Q17: Find the following integrals 17. $\int (2x^2 - 3\sin x + 5\sqrt{x}) dx$
- Q18: Find the following integrals 18. $\int \sec x (\sec x + \tan x) dx$
- Q19: Find the following integrals 19. $\int \frac{\sec^2 x}{\csc^2 x} dx$
- Q20: Find the following integrals 20. $\int \frac{2 - 3\sin x}{\cos^2 x} dx$
- Q21: Choose the correct answer 21. The anti derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals
- Q22: Choose the correct answer 22. If $\frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}$ such that $f(2) = 0$. Then $f(x)$ is
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- Integrals Class 12 EXERCISE 7.4
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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