Download App

Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals4 Marks
Question
Evaluate the following definite integrals:
$\int\limits_{\frac{\pi}{2}}^{\pi}\text{e}^{\text{x}}\Big(\frac{1-\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}$
✓

Answer

Let $\text{I}=\int_{\frac{\pi}{2}}^\limits{\pi}\text{e}^{\text{x}}\Big(\frac{1-\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}$ Then,
$\text{I}=\int_{\frac{\pi}{2}}^\limits{\pi}\text{e}^{\text{x}}\bigg(\frac{1-2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}{2\sin^2\frac{\text{x}}{2}}\bigg)\text{dx}$
$\Rightarrow\text{I}=\int_{\frac{\pi}{2}}^\limits{\pi}\text{e}^{\text{x}}\Big(\frac{1}{2}\text{cosec}^2\frac{\text{x}}{2}-\cot\frac{\text{x}}{2}\Big)\text{dx}$
$\Rightarrow\text{I}=\int_{\frac{\pi}{2}}^\limits{\pi}\frac{1}{2}\text{e}^{\text{x}}\text{cosec}^2\frac{\text{x}}{2}\text{ dx}-\int_{\frac{\pi}{2}}^\limits{\pi}\text{e}^{\text{x}}\cot\frac{\text{x}}{2}\text{ dx}$
Integrating second term by parts,
$\text{I}=\bigg\{-\Big[\text{e}^{\text{x}}\cot\frac{\text{x}}{2}\Big]^{\pi}_\frac{\pi}{2}-\int_{\frac{\pi}{2}}^\limits{\pi}\frac{1}{2}\text{ e}^{\text{x}}\text{cosec}^2\frac{\text{x}}{2}\text{ dx}\bigg\}+\int_{\frac{\pi}{2}}^\limits{\pi}\frac{1}{2}\text{ e}^{\text{x}}\text{cosec}^2\frac{\text{x}}{2}\text{ dx}$
$\Rightarrow\text{I}=-\Big[0-\text{e}^{\frac{\pi}{2}}\Big]$
$\Rightarrow\text{I}=\text{e}^{\frac{\pi}{2}}$

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 "4 Marks" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet? What is the minimum Amount of vitamin A.
→
Differentiate the following functions from first principles:
$\text{e}^{\sqrt{\cot\text{x}}}$
→
Integrate the function $\frac{e^{a \tan ^{-1} x}}{\left(1+x^2\right)^{3 / 2}}$ with respect to $x$.
→
Solve the following differential equation:
$(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}+\text{y}=\tan^{-1}\text{x}$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\cos^5\text{x dx}$
→
Find the general solution of $x \frac{d y}{d x}+2 y=x^{2} \log x$
→
Evaluate the following integrals:
$\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$
→
Verify Rolle's theorem for the following function on the indicated intervals

$\text{f}(\text{x})=\sin2\text{x}\text{ on }\Big[0,\frac{\pi}{2}\Big]$

→
Find the equation of the line passing through the points (-1, 2, 1) and parallel to the line $\frac{2\text{x}-1}{4}=\frac{3\text{y}+5}{4}=\frac{2-\text{z}}{3}.$
→
Differentiate the following functions with respect to x:
$\frac{\text{x}^2+2}{\sqrt{\cos\text{x}}}$
→