Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS2 Marks
Question
Integrate the function in Exercise:$\frac{2+\sin2\text{x}}{1+\cos2\text{x}}\text{e}^{\text{x}}$
✓

Answer

$\text{I}=\int\bigg(\frac{2+\sin2\text{x}}{1+\cos2\text{x}}\bigg)\text{e}^{\text{x}}$
$=\int\bigg(\frac{2+2\sin\text{x}\cos\text{x}}{2\cos^{2}\text{x}}\bigg)\text{e}^{\text{x}}$
$=\int\bigg(\frac{1+\sin\text{x}\cos\text{x}}{\cos^{2}\text{x}}\bigg)\text{e}^{\text{x}}$
$=\int\big(\sec^{2}\text{x}+\tan\text{x}\big)\text{e}^{\text{x}}$
$\text{Let}\ \text{f(x)}=\tan\text{x}\Rightarrow\text{f(x)}=\sec^{2}\text{x}$
$\therefore\text{I}=\int\big[\text{f(x)}+\text{f}'\text{(x)}\big]\text{e}^{\text{x}}\text{dx}$
$=\text{e}^{\text{x}}\ \text{f(x)}+\text{C}$
$=\text{e}^{\text{x}}\tan\text{x}+\text{C}$

Need a full question paper?

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

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

Find the principal value of $\operatorname{cosec}^{-1}(-1)$
→
Determine the direction cosines of the normal to the plane and the distance from the origin.
2x + 3y - z = 5
→
Integrate the rational function $\frac{{2x}}{{\left( {{x^2} + 1} \right)\left( {{x^2} + 3} \right)}}$
→
Write the minimum value of $\text{f}(\text{x})=\text{x}+\frac{1}{\text{x}},\text{x}>0.$
→
Evaluate the following integrals:
$\int\text{xe}^\text{x}\text{dx}$
→
A die is thrown three times. Let X be 'the number of twos seen'. Find the expectation of X.
→
Find $\frac{\text{dy}}{\text{ dx}} $in the following:
$\sin^{2}\text{x}+\cos^{2}\text{y}=1$
→
Prove that:
$\tan^{-1}\sqrt{x}=\frac{1}{2}\cos^{-1}\bigg(\frac{1-x}{1+x}\bigg),x\in\left[0,1\right]$
→
If A is a square matrix satisfying $A^T$ A = l, write the value of |A|.
→
Find the sum of the following vectors: $\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}},\vec{\text{b}}=2\hat{\text{i}}-3\hat{\text{j}},\vec{\text{c}}=2\hat{\text{i}}-3\hat{\text{k}}$.
→