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

Question

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

✓

Answer

$\int\frac{\cos\text{x dx}}{\sin^2\text{x}+4\sin\text{x}+5}$
Let $\sin\text{x = t}$
$\Rightarrow\cos\text{x dx = dt}$
Now, $\int\frac{\cos\text{x dx}}{\sin^2\text{x}+4\sin\text{x}+5}$
$=\int\frac{\text{dt}}{\text{t}^2+4\text{t}+5}$
$=\int\frac{\text{dt}}{\text{t}^2+2\times\text{t}\times2+4+1}$
$=\int\frac{\text{dt}}{(\text{t}+2)^2+1^2}$
$=\frac{1}{1}\tan^{-1}\Big(\frac{\text{t}+2}{1}\Big)+\text{C}$
$=\tan^{-1}(\sin\text{x}+2)+\text{C}$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove the following results
$\tan\Big(\cos^{-1}\frac{4}{5}+\tan^{-1}\frac{2}{3}\Big)=\frac{17}{6}$

The radius of a spherical soap bubble is increasing at the rate of 0.2cm/ sec. Find the rate of increase of its surface area, when the radius is 7cm.

If x and y are connected parametrically by the equation $x = a\left( {\cos t + \log \tan \frac{t}{2}} \right),$ y = a sin t, without eliminating the parameter, find $\frac{{dy}}{{dx}}$.

Integrate the function in Exercise:
$\frac{1}{\text{x}\sqrt{\text{ax}-\text{x}^{2}}}$
$\big[\text{Hint:putx}=\frac{\text{a}}{\text{t}}\big]$

Let $*$ be the binary operation defined on $Q$. Find which of the following binary operations are commutative:

$a * b = a – b \forall a, b \in Q$
$a * b = a^2 + b^2 \forall a, b \in Q$
$a * b = a + ab \forall a, b \in Q$
$a * b = (a – b)^2 \forall a, b \in Q$

For the principal values, evaluate the following:
$\sin^{-1}[\cos\{2\text{cosec}^{-1}(-2)\}]$

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}.$

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=2\text{x}^2+3\sin\text{x}\ \text{at}\text{ x}=0$

Find the value of the following:
$\tan\frac{1}{2}\bigg[\sin^{-1}\frac{2\text{x}}{1 + \text{x}^{2}} + \cos^{-1}\frac{ 1 - \text{y}^{2}}{1 + \text{y}^{2}}\bigg],|\text{x}|<1,\text{y}>0\text{ and } \text{xy} < 1.$

If $\sin^{-1}\frac{2\text{a}}{1+\text{a}^2}-\cos^{-1}\frac{1-\text{b}^2}{1+\text{b}^2}=\tan^{-1}\frac{2\text{x}}{1-\text{x}^2},$ then prove that $\text{x}=\frac{\text{a}-\text{b}}{1+\text{ab}}$