Find the integrals of the functions in Exercises: 2x ( + )^2 - Vidyadip

Question

Find the integrals of the functions in Exercises:
$\frac{\cos2\text{x}}{(\cos\text{x}+\sin\text{x})^2}$

✓

Answer

$\frac{\cos2\text{x}}{(\cos\text{x}+\sin\text{x})^2}=\frac{\cos2\text{x}}{\cos^2\text{x}+\sin^2\text{x}+2\sin\text{x}\cos\text{x}}=\frac{\cos2\text{x}}{1+\sin2\text{x}}$
$\therefore\int\frac{\cos2\text{x}}{(\cos\text{x}+\sin\text{x})^2}\text{ dx}=\int\frac{\cos2\text{x}}{(1+\sin2\text{x})}\text{dx}$
$\text{Let }1+\sin2\text{x}=\text{t}$
$\Rightarrow2\cos2\text{x}\text{ dx}=\text{dt}$
$\therefore\int\frac{\cos2\text{x}}{(\cos\text{x}+\sin\text{x})^2}\text{dx}=\frac{1}{2}\int\frac{1}{\text{t}}\text{dt}$
$=\frac{1}{2}\log|\text{t}|+\text{C}$
$=\frac{1}{2}\log|1+\sin2\text{x}|+\text{C} $
$=\frac{1}{2}\log|(\sin\text{x}+\cos\text{x})^2|+\text{C} $
$=\log|\sin\text{x}+\cos\text{x}|+\text{C}$

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 "3 Marks Question" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Write the vector equation of the line passing through the point (1, -2, -3) and normal to the plane $\vec{\text{r}}.(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})=5.$

$\int\frac{1}{\sqrt{\text{x+a}}+\sqrt{\text{x+b}}}\text{dx}$

Express $\overrightarrow{\text{AB}}$ in terms of unit vectors $\hat{\text{i}}\text{ and }\hat{\text{j}}$, when the point is:A(-6, 3), B(-2, -5)
Find $\Big|\overrightarrow{\text{AB}}\Big|$

Evaluate the following integrals:
$\int\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}\text{dx}$

Integrate the function in Exercise.
$\sqrt{1+\frac{\text{x}^2}{9}}$

$\text{If} \sin [\cot^{-1} ( x + 1)] = \cos(\tan^{-1}x), \text{then find x}.$

If $\text{y}=\text{e}^{\text{a}\cos^{-1}}\text{x}$ prove that $(1-\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-\text{a}^2\text{y}=0$

Find the vector equation of the line passing through the point A(1, 2, -1) and parallel to the line 5x - 25 = 14 - 7y = 35z.

Evaluate the following integrals:
$\int^\limits{9}_4\frac{\sqrt{\text{x}}}{\big(30-\text{x}^{\frac{3}{2}}\big)^2}\text{ dx}$

If $l_1, m_1, n_1$ and $l_2, m_2, n_2$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $m_1n_2 - m_2n_1, l_1n_2 - l_2n_1, l_1m_2 - l_2m_1.$