Download App

INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS5 Marks
Question
Evaluate the following:
$\int\frac{\sin^6\text{x}+\cos^6\text{x}}{\sin^2\text{x}\cos^2\text{x}}\text{dx}$
✓

Answer

Let $\text{I}=\int\frac{\sin^6\text{x}+\cos^6\text{x}}{\sin^2\text{x}\cos^2\text{x}}\text{dx}$ $=\int\frac{(\sin^2\text{x})^3+(\cos^2\text{x})^3}{\sin^2\text{x}\cdot\cos^2\text{x}}\text{dx}$
$=\int\frac{(\sin^2\text{x}+\cos^2\text{x})(\sin^4\text{x}-\sin^2\text{x}\cos^2\text{x}+\cos^4\text{x})}{\sin^2\text{x}\cdot\cos^2\text{x}}\text{dx}$
$=\int\frac{\sin^4\text{x}}{\sin^2\text{x}\cos^2\text{x}}\text{dx}+\int\frac{\cos^4\text{x}}{\sin^2\text{x}\cdot\cos^2\text{x}}\text{dx}-\int\frac{\sin^2\text{x}\cos^2\text{x}}{\sin^2\text{x}\cdot\cos^2\text{x}}\text{dx}$
$=\int\tan^2\text{xdx}+\int\cos^2\text{xdx}-\int1\text{dx}$
$=\int(\sec^2\text{x}-1)\text{dx}+\int(\cos\text{ec}^2\text{x}-1)\text{dx}-\int1\text{dx}$
$=\int\sec^2\text{xdx}+\int\cos\text{ec}^2\text{xdx}-3\int\text{dx}$
$\text{I}=\tan\text{x}-\cot\text{x}-3\text{x}+\text{C}$

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 "5 Marks Questions" 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

Find the equation of the plane which is perpendicular to the plane $5x + 3y + 6z + 8 = 0$ and which contains the line of intersection of the planes $x + 2y + 3z - 4 = 0$ and $2x + y - z + 5 =0$
→
Evaluate the following integrals:
$\int(\tan^{-1}\text{x}^2)\text{x dx}$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix}2 & 5 \\ 1 & 3 \end{bmatrix}$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\frac{\cos^{2}\text{x}}{1+\text{e}^{\text{x}}}\text{ dx}$
→
Evaluate the following integrals:
$\int\limits^{1}_0\frac{\log(1+\text{x})}{1+\text{x}^2}\text{ dx}$
→
Evaluate the following integrals:
$\int_{1}^\limits{2}\frac{1}{\text{x}\big(1+\log\text{x}\big)^2}\text{ dx}$
→
Evaluate the following integral:
$\int\frac{\text{x}^4+1}{\text{x}^2+1}\text{ dx}$
→
If $x = a \cos^3 \theta$ and $y = a \sin^3 \theta$, then find the value of $\frac{\text{f}^{2}\text{y}}{\text{dx}}\text{at}\theta = \frac{\pi}{6}.$
→
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = (\text{x}+\text{y})^2$
→