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Q15:
Find the following integrals 15. $\int \sqrt{x} (3x^2 + 2x + 3) dx$
Find the following integrals 15. $\int \sqrt{x} (3x^2 + 2x + 3) dx$
Solution :
Step 1: Expanding the Integrand
The given integral is: $I = \int \sqrt{x} (3x^2 + 2x + 3) \, dx$
We express $\sqrt{x}$ as a power of $x$, specifically $x^{1/2}$. Applying the Distributive Property of Multiplication over Addition [Per the Distributive Law of Arithmetic]:
$I = \int x^{1/2} (3x^2 + 2x + 3) \, dx$
$I = \int (3x^{2 + 1/2} + 2x^{1 + 1/2} + 3x^{1/2}) \, dx$
$I = \int (3x^{5/2} + 2x^{3/2} + 3x^{1/2}) \, dx$
Step 2: Applying the Linearity Property of Integrals
According to the Linearity of the Integral operator, the integral of a sum is the sum of the integrals [Per the Sum Rule for Integration]:
$I = 3 \int x^{5/2} \, dx + 2 \int x^{3/2} \, dx + 3 \int x^{1/2} \, dx$
Step 3: Calculating Antiderivatives
We apply the Power Rule for integration, which states $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, provided $n \neq -1$ [Per the Power Rule of Calculus].
- For $\int x^{5/2} \, dx$: $\frac{x^{5/2 + 1}}{5/2 + 1} = \frac{x^{7/2}}{7/2} = \frac{2}{7}x^{7/2}$
- For $\int x^{3/2} \, dx$: $\frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5}x^{5/2}$
- For $\int x^{1/2} \, dx$: $\frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}$
Step 4: Final Assembly and Simplification
Substituting the results back into the expression:
$I = 3 \left( \frac{2}{7} x^{7/2} \right) + 2 \left( \frac{2}{5} x^{5/2} \right) + 3 \left( \frac{2}{3} x^{3/2} \right) + C$
$I = \frac{6}{7} x^{7/2} + \frac{4}{5} x^{5/2} + 2 x^{3/2} + C$
Final Solution: $\frac{6}{7} x^{7/2} + \frac{4}{5} x^{5/2} + 2 x^{3/2} + C$
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. $\sin 2x$
- Q2: Find an anti derivative (or integral) of the following functions by the method of inspection. 2. $\cos 3x$
- 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. $\int (1 - x)\sqrt{x} dx$
- 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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