Evaluate the definite integral in Exercise: ^ _ 0 ( ^ 2 x 2 - ^ 2… - Vidyadip

Question

Evaluate the definite integral in Exercise:
$\int^{\pi}_{0}(\sin^{2}\frac{\text{x}}{2}-\cos^{2}\frac{\text{x}}{2})\text{dx}$

✓

Answer

$\text{Let}\text{I}=\int\limits_{0}^{\pi}\bigg(\sin^{2}\frac{\text{x}}{2}-\cos^{2}\frac{\text{x}}{2}\bigg)\text{dx}$$=-\int\limits_{0}^{\pi}\bigg(\cos^{2}\frac{\text{x}}{2}-\sin^{2}\frac{\text{x}}{2}\bigg)\text{dx}$
$=-\int\limits_{0}^{\pi}\cos\text{x}\ \text{dx}$
$\int\cos\text{x}\text{dx}=\sin\text{x}=\text{F}\text{(x)}$
By second fundamental theorem of calculus, we obtain
$\text{I}=\text{F}(\pi)-\text{F}(0)$
$=\sin\pi-\sin0$
$=0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Fatima and John appear in an interview for two vacancies for the same post. The probability of Fatima's selection is $\frac{1}{7}$ and that of John's selection is $\frac{1}{5}$. What is the probability that,
None of them will be selected?

Find the angle between the vectors $\vec{\text{a}} $ and $\vec{\text{b}},$ where$\vec{\text{a}}=2\hat {\text{i}}-\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}} =4\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$

Determine whether the following operations define a binary operation on the given set or not:
'+6' on $S = \{0, 1, 2, 3, 4, 5\}$ defined by, $\text{a}+_6\text{b}=\begin{cases}\text{a}+\text{b},&\text{if a}+\text{b}<6\\\text{a}+\text{b}-6,&\text{if a}+\text{b}\geq6\end{cases}$

Six coins are tossed simultaneously. Find the probability of getting.
3 heads.

Write the domain of the real function $\text{f(x)}=\sqrt{\text{x}-[\text{x}]}.$

If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are non-coplanar vectors, then find the value of $\frac{\vec{\text{a}}.\big(\vec{\text{b}}\times\vec{\text{c}}\big)}{\big(\vec{\text{c}}\times\vec{\text{a}}\big).\vec{\text{b}}}+\frac{\vec{\text{b}}.\big(\vec{\text{a}}\times\vec{\text{c}}\big)}{\vec{\text{c}}.\big(\vec{\text{a}}\times\vec{\text{b}}\big)}$

Differentiate $(x^2 - 5x + 8) (x^3 + 7x + 9)$ in three ways mentioned below:

by using product rule
by expanding the product to obtain a single polynomial
by logarithmic differentiation.
Do they all give the same answer?

Show that the following triads of vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}},\vec{\text{b}}=3\hat{\text{i}}+2\hat{\text{j}}+7\hat{\text{k}},\vec{\text{c}}=5\hat{\text{i}}+6\hat{\text{j}}+5\hat{\text{k}}$

If A is a singular matrix, then write the value of |A|.

Find the components along the coordinate axis of the position vector of the following point:
P(3, 2)