Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals:$\int\limits^{\pi}_0\text{x}\log\sin\text{x}\text{ dx}$
✓

Answer

$\int\limits^{\pi}_0\text{x}\log\sin\text{x}\text{ dx}$Let $\text{I}=\int\limits^{\pi}_0\text{x}\log\sin\text{x}\text{ dx}\ ...(\text{i})$
$\text{I}=\int\limits^{\pi}_0(\pi-\text{x})\log\sin(\pi-\text{x})\text{dx}$
$\text{I}=\int\limits^{\pi}_0(\pi-\text{x})\log(\sin\text{x})\text{dx}\ ...(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\pi\int\limits^{\pi}_0\log\sin\text{x}\text{ dx}$
$=2\pi\int\limits^{\frac{\pi}{2}}_0\log\sin\text{x}\text{ dx}$
$\text{I}=\pi\int\limits^{\frac{\pi}{2}}_0\log\sin\text{x}\text{ dx}\ ....(\text{iii})$
Let $\int\limits^{\frac{\pi}{2}}_0\log\sin\text{x}\text{ dx}=\text{I}_2$
$\text{I}_2=\int\limits^{\frac{\pi}{2}}_0\log\sin\Big(\frac{\pi}{2}-\text{x}\Big)\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log\cos\text{x dx}$
$2\text{I}_2=\int\limits^{\frac{\pi}{2}}_0(\log\sin\text{x}+\log\cos\text{x})\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log(\sin\text{x}\cos\text{x})\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log(\sin2\text{x})\text{dx}-\int\limits^{\frac{\pi}{2}}_0\log2\text{ dx}$
Let $2\text{x}=\text{t}$
$2\text{dx}=\text{dt}$
When, $\text{x}=0\Rightarrow\text{t}=0$
$\text{x}=0\Rightarrow\text{t}=\pi$
$2\text{I}_2=\frac{1}{2}\int\limits^{{\pi}}_0\log(\sin\text{t})\text{dt}-\frac{\pi}{2} \log2 $
$2\text{I}_2=\frac{2}{2}\int\limits^{{\pi}}_0\log(\sin\text{t})\text{dt}-\frac{\pi}{2} \log2 $
$2\text{I}_2=\text{I}_2-\frac{\pi}{2} \log2 $
$\text{I}_2=-\frac{\pi}{2}\log2$
From (iii),
$\text{I}=\pi\int\limits^{{\pi}}_0\log\sin\text{x}\text{dx}=\pi\text{I}_2$
$\text{I}=\pi\Big(-\frac{\pi}{2}\log2\Big)$
$\text{I}=\frac{-\pi^2\log2}{2}$

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 "Solve the Following Question.(5 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following integral:
$\int\frac{1}{\sqrt{(2-\text{x})^2+1}}\text{ dx}$
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}+1}{(\text{x}+1)^2(\text{x}+2)}\ \text{dx}$
→
Show that the four points A, B, C, D with position vectors $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}},\ \vec{\text{d}}$ respectively such that $3\vec{\text{a}}-2\vec{\text{b}}+5\vec{\text{c}}-6\vec{\text{d}}=0$, are coplanar. Also, find the position vector of the point of intersection of the line segments AC and BD.
→
Evaluvate the following intregals
$\int\frac{1}{4+3\tan\text{x}}\ \text{dx}$
→
Find $\lambda$ for which the points A(3, 2, 1), B(4, $\lambda$, 5), C(4, 2, -2) and D(6, 5, -1) are coplanar.
→
If $(\text{x}-\text{y})\text{e}^{\frac{\text{x}}{\text{x}-\text{y}}}=\text{a},$ prove that $\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}=2\text{y}$
→
If $\text{y}=\sqrt{\text{x}^2+\text{a}^2},$ prvoe that $\text{y}\frac{\text{dy}}{\text{dx}}-\text{x}=0$
→
If $\text{y}=(\sin^{-1}\text{x})^2,$ prove that $(1-\text{x}^2)\text{y}_2-\text{xy}_1-2=0$
→
Find $A^{-1},$ If $\text{A}=\begin{bmatrix}1&2&5\\ 1&-1&-1\\ 2&3&-1\end{bmatrix}$. Hence solve the follwing system of linear equations:
$x + 2y +5z = 10, x- y - z = - 2, 2x + 3y - z = - 11$
→
Show that the points whose position vectors are $-2\hat{\text{i}}+3\hat{\text{j}},\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$ and $7\hat{\text{i}}-\hat{\text{k}}$ are collinear.
→