Question
Evaluate the following definite integrals:$\int_{0}^\limits{\pi}\Big(\sin^2\frac{\text{x}}{2}-\cos^2\frac{\text{x}}{2}\Big)\text{dx}$
✓
Answer
Let $\text{I}=\int_{0}^\limits{\pi}\Big(\sin^2\frac{\text{x}}{2}-\cos^2\frac{\text{x}}{2}\Big)\text{dx}$$=-\int_{0}^\limits{\pi}\Big(\cos^2\frac{\text{x}}{2}-\sin^2\frac{\text{x}}{2}\big)\text{dx}$
$=-\int_{0}^\limits{\pi}\cos\text{x dx}$
$\int\cos\text{x dx}=\sin\text{x}=\text{F(x)}$
By second fundamental theorem of calculus, we obtain$\text{I}=\text{F}(\pi)-\text{F}(0)$
$=\sin(\pi)-\sin0$
$=0$
$=-\int_{0}^\limits{\pi}\cos\text{x dx}$
$\int\cos\text{x dx}=\sin\text{x}=\text{F(x)}$
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.
Explore more
Similar questions
$\int\cos^42\text{x dx}$
→How many times must a man toss a fair coin so that the probability of having at least one head is more than 80%?
Evaluate the following definite integrals:$\int_{-1}^\limits{1}\frac{1}{\text{x}^2+2\text{x}+5}\text{ dx}$
→Evaluate the following intregals:
$\int\frac{1}{\cos2\text{x}+3\sin^2\text{x}}\ \text{dx}$
→$\int\frac{1}{\cos2\text{x}+3\sin^2\text{x}}\ \text{dx}$
Diffrentiate the following w.r.t.x
→$\frac{1+\sin x^{\circ}}{1-\sin x^{\circ}}$
Evaluate the following integrals:
$\frac{\text{x}+1}{\text{x}(1+\text{xe}^\text{x})}\ \text{dx}$
→$\frac{\text{x}+1}{\text{x}(1+\text{xe}^\text{x})}\ \text{dx}$
If $\mathrm{D}, \mathrm{E}, \mathrm{F}$ are the mid-points of the sides $\mathrm{BC}, \mathrm{CA}, \mathrm{AB}$ of a triangle $\mathrm{ABC}$, prove that $\overline{A D}+$
→$\overline{B E}+\overline{C F}=0$
ABCD is a parallelogram and P is the point of intersection of its diagonals. If O is the origin of reference, show that $\overrightarrow{\text{OA}}+\overrightarrow{\text{OB}}+\overrightarrow{\text{OC}}+\overrightarrow{\text{OD}}=4\ \overrightarrow{\text{OP}}$.
→Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
If $A^{-1}=\frac{1}{3}\left[\begin{array}{ccc}1 & 4 & -2 \\ -2 & -5 & 4 \\ 1 & -2 & 1\end{array}\right]$ and $|A|=3$, then (adj. $\left.A\right)=\ldots$.
→