Evaluate the following intregals: 1 1+3 ^2x dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{1+3\sin^2\text{x}}\text{ dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{1+3\sin^2\text{x}}\text{ dx}$
Dividing numerator and denominator by $\cos^2\text{x}$
$\text{I}=\int\frac{\frac{1}{\cos^2\text{x}}}{\frac{1}{\cos^2\text{x}}+\frac{3\sin^2\text{x}}{\cos^2\text{x}}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{\sec^2\text{x}+3\tan^2\text{x}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{1+\tan^2\text{x}+3\tan^2\text{x}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{1+4\tan^2\text{x}}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{1+(2\tan\text{x})}\ \text{dx}$
Let $2\tan\text{x}=\text{t}$
$2\sec^2\times\text{dx}=\text{dt}$
$\text{I}=\frac{1}{2}\int\frac{\text{dt}}{1+\text{t}^2}$
$=\frac{1}{2}\tan^{-1}\text{t}+\text{c}$
$\text{I}=\frac{1}{2}\tan^{-1}(2\tan\text{x})+\text{C}$

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 "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

Evaluate the following integrals:
$\int(2\text{x}-5)\sqrt{2+3\text{x}-\text{x}^2}\text{dx}$

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$

ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD, BC and AD. Show that $\overrightarrow{\text{PA}}+\overrightarrow{\text{PB}}+\overrightarrow{\text{PC}}+\overrightarrow{\text{PD}}=4\ \overrightarrow{\text{PQ}}$, where P ia any point.

If the interest is compounded continuously at $6\%$ per annum, how much worth $Rs.100$ will be after $10$ years? How long will it take to double $Rs. 1000?$

Evaluate the following integrals: $\int\frac{1}{\cos\text{x}(5-4\sin\text{x})}\ \text{dx}$

Differentiate the following functions with respect to x:
$\sqrt{\frac{1+\sin\text{x}}{1-\sin\text{x}}}$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\log(\text{x}^2+2)-\log3\text{ on }[-1,1]$

A beam is supported at the two ends and is uniformly loaded. The bending moment M at a distance x from one end is given by
$\text{M}=\frac{\text{WL}}{2}\text{x}-\frac{\text{W}}{3}\frac{\text{x}^{3}}{\text{L}^{2}}$
Find the point at which M is maximum in each case.

Evaluate the following intregals:
$\int\frac{1}{4\sin^2\text{x}+5\cos^2\text{x}}\text{ dx}$

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point $'c\ '$ in the indicated interval as stated by the Lagrange's mean value theorem. $f(x) = 2x - x^{2}$ on $[0, 1]$