Using properties of definite integrals, evaluate: ^ /4 _ 0 log (1… - Vidyadip

Question

Using properties of definite integrals, evaluate:
$\int\limits^{\pi/4}_{0}\text{log (1 + tan x) dx}$.

✓

Answer

$\text{I}=\int_{0}^{\pi/4}\log\text{(1 + tan x ) dx}=\int_0^{\pi/4}\log\Bigg(1+\tan(\frac{\pi}{4}-\text{x})\Bigg)\text{dx}$
$=\int_{0}^{\pi/4}\log\Bigg(1+\frac{1-\tan\text{x}}{\text{1 + tan x}}\Bigg)\text{dx}=\int_{0}^{\pi/4}\log\Bigg(\frac{2}{\text{1 + tan x}}\Bigg)\text{dx}$

$=\int_{0}^{\pi/4}\log2\text{ dx}-\int_0^{\pi/4}\log(1+\tan\text{ x})\text{ dx}$

$\Rightarrow\text{2I}=\log2\cdot\int_0^{\pi/4}1\cdot\text{dx}=\log2\cdot\pi/4$

$\Rightarrow\text{I}=\pi/8\cdot\log2.$

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

For the matrices $A$ and $B$, verify that $(AB)^T = B^TA^T$, where $\text{A}=\begin{bmatrix}1&3\\2&4\end{bmatrix},\text{B}=\begin{bmatrix}1&4\\2&5\end{bmatrix}$

Let C be the set of all complex numbers and $C_0$ be the set of all no-zero complex numbers. Let a relation R on $C_0$ be defined as $\text{z}_1\text{R z}_2\Leftrightarrow\frac{\text{z}_1-\text{z}_2}{\text{z}_1+\text{z}_2}$ is real for all $\text{z}_1,\ \text{z}_2\in\text{C}_0.$ Show that R is an equivalence relation.

Verify Rolle's theorem for the following function on the indicated intervals
$f(x) = x^2 -4x + 3$ on $[1, 3]$

Solve the following differential equation:
$(\text{x}^2-2\text{xy})\text{dy}+(\text{x}^2-3\text{xy}+2\text{y}^2)\text{dx}=0$

Solve the following differential equations:$\text{x}\sqrt{1-\text{y}^2}\text{dx}+\text{y}\sqrt{1-\text{x}^2}\text{dy}=0$

If $\text{x}=\Big(\text{t}+\frac{1}{\text{t}}\Big)^\text{a},\text{y}=\text{a}^{\text{t}+\frac{1}{\text{t}}},$ find $\frac{\text{dy}}{\text{dx}}$

Find: $\int \frac{x \sin^{-1} x}{\sqrt{1 - x^{2}}} \text{d}x.$

Show that the following set of curves intersect orthogonally.
$y = x^3$ and $6y = 7 - x^2$

By Using properties of definite integrals, evaluate the following integral in Exercise:
$\int^{\pi}_{0}\log(1+\cos\text{x})\text{dx}$

Using properties of determinants, prove the following:
$\begin{vmatrix} 1 & \text{1 + P} & \text{1 + p + q} \\ 2 & \text{3 + 2p} & \text{1 + 3p + 2q} \\ 3 & \text{6 + 3p} & \text{1 + 6p + 3q} \end{vmatrix}=1.$