Evaluate the definite integral: \(∫^\frac{π}{2}_0 cos^2 x dx\)
Evaluate the definite integral: \(∫^\frac{π}{2}_0 cos^2 x dx\)
Solution and Explanation
Let I=\(∫^\frac{π}{2}_0 cos^2 x dx\)
\(∫cos^2 x dx=∫(\frac{1+cos2x}{2})dx=\frac{x}{2}+\frac{sin2x}{4}=\frac{1}{2}(x+\frac{sin2x}{2})=F(x)\)
By second fundamental theorem of calculus,we obtain
\(I=[F(\frac{π}{2})-F(0)]\)
\(=\frac{1}{2}[(\frac{π}{2})-\frac{sinπ}{2})-(0+\frac{sin0}{2})]\)
\(=\frac{1}{2}[\frac{π}{2}+0-0-0]\)
\(=\frac{π}{4}\)
Top Questions on integral
- Evaluate the integral: $$ \int_0^{\pi/4} \frac{\ln(1 + \tan x)}{\cos x \sin x} \, dx $$
- Find the value of the integral: \[ \int_0^\pi \sin^2(x) \, dx. \]
- Evaluate the integral \( \int_0^1 \frac{\ln(1 + x)}{1 + x^2} \, dx \)
- Evaluate the integral: $$ \int_0^1 \frac{\ln (1+x)}{1+x^2} \, dx $$
- Evaluate $ \int_{0}^{1} (3x^2 + 2x) \, dx $.
Questions Asked in CBSE CLASS XII exam
- Explain any two makeup removal techniques.
- CBSE CLASS XII - 2025
- Makeup Techniques and Tools
- Describe in detail, as to how to book an appointment in the salon.
- CBSE CLASS XII - 2025
- Basic Salon Management Skills
- Describe the procedure of lymphatic drainage facial and its contraindications.
- CBSE CLASS XII - 2025
- facial equipment
- How will you provide a caring environment to your client to make her feel comfortable? Explain.
- CBSE CLASS XII - 2025
- Basic Salon Management Skills
- Explain the importance of storage area.
- CBSE CLASS XII - 2025
- Basic Salon Management Skills
Concepts Used:
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus is the theorem which states that differentiation and integration are opposite processes (or operations) of one another.
Calculus's fundamental theorem connects the notions of differentiating and integrating functions. The first portion of the theorem - the first fundamental theorem of calculus – asserts that by integrating f with a variable bound of integration, one of the antiderivatives (also known as an indefinite integral) of a function f, say F, can be derived. This implies the occurrence of antiderivatives for continuous functions.