Evaluate the following integrals: ^ 2 _ 0 1+m^2 ^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}\frac{\tan\text{x}}{1+\text{m}^2\tan^2\text{x}}\text{ dx}$

✓

Answer

Let $\text{I}=\int^\limits{\frac{\pi}{2}}_{0}\frac{\tan\text{x}}{1+\text{m}^2\tan^2\text{x}}\text{ dx}$
$=\int^\limits{\frac{\pi}{2}}_{0}\frac{\frac{\sin\text{x}}{\cos\text{x}}}{1+\text{m}^2\frac{\sin^2\text{x}}{\cos^2\text{x}}}\text{ dx}=\int^\limits{\frac{\pi}{2}}_{0}\frac{{\sin\text{x}}{\cos\text{x}}}{\cos^2\text{x}+\text{m}^2{\sin^2\text{x}}}\text{ dx}$
Put $\cos^2\text{x}+\text{m}^2\sin^2\text{x}=\text{z}$
$\therefore\ 2\cos\text{x}(-\sin\text{x})\text{dx}+\text{m}^2\times2\sin\text{x}\cos\text{x dx}=\text{ dz}$
$\Rightarrow2(\text{m}^2-1)\sin\text{x}\cos\text{x dx}=\text{dz}$
$\Rightarrow\sin\text{x}\cos\text{x dx}=\frac{\text{dz}}{2(\text{m}^2-1)}$
When $\text{x}\rightarrow0,\text{ z}\rightarrow1$ $\big(\text{z}=\cos^2\text{x}+\text{m}^2\sin^2\text{x}=1+\text{m}^2\times0=1\big)$
When $\text{x}\rightarrow\frac{\pi}{2},\text{ z}\rightarrow\text{m} ^2$ $\big(\text{z}=\cos^2\text{x}+\text{m}^2\sin^2\text{x}=0+\text{m}^2\times0=\text{m}^2\big)$
$\therefore\ \text{I}=-\frac{1}{2(\text{m}^2-1)}\int\limits^{\text{m}^2}_1\frac{\text{dz}}{\text{z}}$
$=\frac{1}{2(\text{m}^2-1)}\big[\log\text{z}\big]^{\text{m}^2}_1$
$=\frac{1}{2(\text{m}^2-1)}\big(\log\text{m}^2-\log1\big)$
$=\frac{1}{2(\text{m}^2-1)}\big(2\log|\text{m}|-0\big)$
$=\frac{\log|\text{m}|}{\text{m}^2-1}$

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

Similar questions

Show that the following system of linear equation is inconsistent:
$3x − y − 2z = 2$
$2y − z = −1$
$3x − 5y = 3$

If sin y = x sin (a + y), prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^{2}\text{(a + y)}}{\sin\text{a}}.$

If $\lim\limits_{\text{x}\rightarrow{\text{c}}}\frac{\text{f(x)}-\text{f(c)}}{\text{x}-\text{c}}$ exists finitely, write the value of $\lim\limits_{\text{x}\rightarrow{\text{c}}}\text{f(x)}.$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=1+\text{x}+\text{y}^2+\text{xy}^2$ when $\text{y}=0,\text{x}=0$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = (\text{x}+\text{y})^2$

Two schools $P$ and $Q$ want to award their selected students on the values of Discipline, Politeness and Punctuality. The school $P$ wants to award ₹ $x$ each, ₹ $y$ each and ₹ $z$ each for the three respective values to its $3, 2$ and $1$ students with a total award money of ₹ $1,000.$ School $Q$ wants to spend ₹ $1,500$ to award its $4, 1$ and $3$ students on the respective values $($by giving the same award money for the three values as before$).$ If the total amount of awards for one prize one each value is ₹ $600,$ using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

Find the equation of a curve passing through the point $(0, 2)$ given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by $5.$

Find the value of $4\tan^{-1}\frac{1}{5}-\tan^{-1}\frac{1}{239}.$

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 3 & -1 \\ 1 & -1 & 3 \end{bmatrix}$

Show that the following set of curves intersect orthogonally.
$x^3 - 3xy^2 = -2$ and $3x^2y - y^3 = 2$