Question
Evaluate the following integrals:$\int\frac{\text{x}+\sin\text{x}}{1+\cos\text{x}}\text{dx}$
✓
Answer
$\int\Big(\frac{\text{x}+\sin\text{x}}{1+\cos\text{x}}\Big)\text{dx}$
$=\int\Big[\frac{\text{x}}{1+\cos\text{x}}+\frac{\sin\text{x}}{1+\cos\text{x}}\Big]\text{dx}$
$=\int\bigg[\frac{\text{x}}{2\cos^{2}\frac{\text{x}}{2}}+\frac{2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}{2\cos^2\frac{\text{x}}{2}}\bigg]\text{dx}$
$=\frac{1}{2}\int\text{x}\cdot\sec^2\frac{\text{x}}{2}\text{dx}+\int\tan\frac{\text{x}}{2}\text{dx}$
$=\frac{1}{2}\bigg[\text{x}\cdot\frac{\tan\big(\frac{\text{x}}{2}\big)}{\frac{1}{2}}-\int1\times2\tan\big(\frac{\text{x}}{2}\big)\text{dx}\bigg]+\frac{\log\big|\sec\frac{\text{x}}{2}\big|}{\frac{1}{2}}+\text{C}$
$=\text{x}\tan\big(\frac{\text{x}}{2}\big)-\frac{\log\big|\sec\frac{\text{x}}{2}\big|}{\frac{1}{2}}+\log\frac{\big|\sec\frac{\text{x}}{2}\big|}{\frac{1}{2}}+\text{C}$
$=\text{x}\tan\big(\frac{\text{x}}{2}\big)+\text{C}$
$=\int\Big[\frac{\text{x}}{1+\cos\text{x}}+\frac{\sin\text{x}}{1+\cos\text{x}}\Big]\text{dx}$
$=\int\bigg[\frac{\text{x}}{2\cos^{2}\frac{\text{x}}{2}}+\frac{2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}{2\cos^2\frac{\text{x}}{2}}\bigg]\text{dx}$
$=\frac{1}{2}\int\text{x}\cdot\sec^2\frac{\text{x}}{2}\text{dx}+\int\tan\frac{\text{x}}{2}\text{dx}$
$=\frac{1}{2}\bigg[\text{x}\cdot\frac{\tan\big(\frac{\text{x}}{2}\big)}{\frac{1}{2}}-\int1\times2\tan\big(\frac{\text{x}}{2}\big)\text{dx}\bigg]+\frac{\log\big|\sec\frac{\text{x}}{2}\big|}{\frac{1}{2}}+\text{C}$
$=\text{x}\tan\big(\frac{\text{x}}{2}\big)-\frac{\log\big|\sec\frac{\text{x}}{2}\big|}{\frac{1}{2}}+\log\frac{\big|\sec\frac{\text{x}}{2}\big|}{\frac{1}{2}}+\text{C}$
$=\text{x}\tan\big(\frac{\text{x}}{2}\big)+\text{C}$
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
If $\text{f}\text{(x)}=\begin{cases}\frac{\sin3\text{x}}{\text{x}},& \text{when}\text{ x}\neq0 \\1,&\text{when} \text{ x}=0\end{cases}$ Find whether f(x) is continuous at x = 0.
→Find the angle between the given planes.
$\vec{\text{r}}\cdot(2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}})=1$ and $\vec{\text{r}}\cdot(-\hat{\text{i}}+\hat{\text{j}})=4$
→$\vec{\text{r}}\cdot(2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}})=1$ and $\vec{\text{r}}\cdot(-\hat{\text{i}}+\hat{\text{j}})=4$
Evaluate the following integrals:
$\int\text{x}\sin2\text{x dx}$
→$\int\text{x}\sin2\text{x dx}$
The feasible region for a LPP is shown in. Find the minimum value of Z = 11x + 7y.
If a unit vector $\vec{\text{a}}$ makes an angle $\frac{\pi}{3}$ with $\hat{\text{i}},\frac{\pi}{4}$ with $\hat{\text{j}}$ and an acute angle $\theta$ with $\hat{\text{k}}$, and ,then find the value of $\theta$.
→The cartesian equation of a line AB are $\frac{2\text{x}-1}{\sqrt{3}}=\frac{\text{y}+2}{2}=\frac{\text{z}-3}{3}.$ Find the direction cosines of a line parallel to AB.
→Integrate the rational function $\frac{x^{3}+x+1}{x^{2}-1}$
→Evaluate the following integrals:
$\int^\limits{\pi}_{0}5\big(5-4\cos\theta\big)^{\frac{1}{4}}\sin\theta\text{ d}\theta$
→$\int^\limits{\pi}_{0}5\big(5-4\cos\theta\big)^{\frac{1}{4}}\sin\theta\text{ d}\theta$
Using determinants show that the following points are collinear:
$(5, 5), (-5, 1)$ and $(10, 7)$
→$(5, 5), (-5, 1)$ and $(10, 7)$
Evaluate the following integrals:$\int\frac{\text{x}+7}{3\text{x}^2+25\text{x}+28}\text{ dx}$
→