Evaluate the following intregals: 1 4 ^2x+5 ^2x dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int\frac{1}{4\sin^2\text{x}+5\cos^2\text{x}}\text{ dx}$
Dividing numerator and denominator by $\cos^2\text{x}$
$\text{I}=\int\frac{\frac{1}{\cos^2\text{x}}}{4\tan^2\text{x}+5}\ \text{dx}$
$=\int\frac{\sec^2\text{x}}{4\tan^2\text{x}+5}\ \text{dx}$
Let $\tan\text{x}=\text{t}$
$\sec^2\text{x}\ \text{dx}=\text{dt}$
$\text{I}=\int\frac{\text{dt}}{4+9(\text{t})^2}$
$=\int\frac{\text{dt}}{4\text{t}^2+5}$
Let $2\text{t}=\text{u}$
$2\text{dt}=\text{du}$
$\text{I}=\frac{1}{2}\int\frac{\text{du}}{(4)^2+(\sqrt{5})^2}$
$=\frac{1}{2}\times\frac{1}{\sqrt{5}}\times\tan^{-1}\Big(\frac{\text{u}}{\sqrt{5}}\Big)+\text{C}$
$=\frac{1}{2\sqrt{5}}\tan^{-1}\Big(\frac{2\text{t}}{\sqrt{5}}\Big)+\text{C}$
$\text{I}=\frac{1}{2\sqrt{5}}\tan^{-1}\Big(\frac{2\tan\text{x}}{\sqrt{5}}\Big)+\text{C}$

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 "4 Marks" from Indefinite Integrals→Indefinite 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 length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{\text{r}}.\big(\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}}\big)+5=0.$

Solve the following differential equations:
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$

If $\text{x}=\sin^{-1}\Big(\frac{2\text{t}}{1+\text{t}^2}\Big)$ and $\text{y}=\tan^{-1}\Big(\frac{2\text{t}}{1-\text{t}^2}\Big),-1<\text{t}<1,$ prove that $\frac{\text{dy}}{\text{dx}}=1$

Maximize Z = 5x + 3y
Subject to
$3\text{x}+5\text{y}\leq15$
$5\text{x}+2\text{y}\leq10$
$\text{x},\text{y}\geq0$

Using properties of determinant prove that$
\left|\begin{array}{ccc}
1 & 1+p & 1+p+q \\
2 & 3+2 p & 4+3 p+2 q \\
3 & 6+3 p & 10+6 p+3 q
\end{array}\right|=1
$

Differentiate the function $x^{x \cos x}+\frac{x^{2}+1}{x^{2}-1}$ w.r.t. x.

Prove the following results:
$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{4}}\sec\text{x}\text{ dx}$

Let $\text{A}=\{\text{x}\in\text{Z}:0\leq\text{x}\leq12\}.$ Show that, $\text{R}=\{(\text{a, b}):\text{a, b}\in\text{A},\ |\text{a}-\text{b}|$ is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].

Solve the following differential equations:

$(\text{xy}^2+2\text{x})\text{dx}+(\text{x}^2\text{y+2y})\text{dy}=0$