Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following definite integrals:$\int_{0}^\limits{\frac{\pi}{4}}\sec\text{x}\text{ dx}$
✓

Answer

Let $\int_{0}^\limits{\frac{\pi}{4}}\sec\text{x}\text{ dx}$ Then,$\text{I}=\int_{0}^\limits{\frac{\pi}{4}}\sec\text{x}\frac{\sec\text{x}+\tan\text{x}}{\sec\text{x}+\tan\text{x}}\text{ dx}$
$\Rightarrow\text{I}=\int_{0}^\limits{\frac{\pi}{4}}\frac{\sec^2\text{x}+\sec\text{x}\tan\text{x}}{\sec\text{x}+\tan\text{x}}\text{ dx}$
Put $\text{u}=\sec\text{x}+\tan\text{x}$$\Rightarrow\text{du}=\sec^2\text{x}+\sec\text{x}\tan\text{x dx}$
$\therefore\ \int_{0}^\limits{\frac{\pi}{4}}\frac{\sec^2\text{x}+\sec\text{x}\tan\text{x}}{\sec\text{x}+\tan\text{x}}\text{ dx}=\int\frac{\text{du}}{\text{u}}$
$\Rightarrow\text{I}=\big[\log\text{u}\big]$
$\Rightarrow\text{I}=\big[\log(\sec\text{x}+\tan\text{x})\big]^{\frac{\pi}{4}}_0$
$\Rightarrow\text{I}=\log\Big(\sec\frac{\pi}{4}+\tan\frac{\pi}{4}\Big)-\log(\sec0+\tan0)$
$\Rightarrow\text{I}=\log\big(\sqrt{2}+1\big)-\log1$
$\Rightarrow\text{I}=\log\big(\sqrt{2}+1\big)$

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 "Solve the Following Question.(5 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Using differentials, find the approximate values of the following:
$\sqrt{37}$
→
Find $\frac{\text{dy}}{\text{dx}}$
$y = x^n + n^x + x^x + n^n$​​​​​​​
→
Find the equation of the tangent to the curve $\text{y}=\sqrt{3}\text{x}-2$ which is parallel to the 4x − 2y + 5 = 0.
→
If the straight line $\text{x}\cos\alpha+\text{y}\sin\alpha=\text{p}$ touches the curve $\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1,$ then prove that $\text{a}\cos^2\alpha-\text{b}^2\sin^2\alpha=\text{p}^2.$
→
Find the cartesian and vector equations of a line which passes through the point ($1, 2, 3$) and is parallel to the line $\frac{-\text{x}-2}{1}=\frac{\text{y}+3}{7}=\frac{2\text{z}-6}{3}.$
→
Find the points on the curve $xy + 4 = 0$ at which the tangents are inclined at an angle of $45^\circ $ with the x-axis.
→
Show that the function $\text{f(x)}\begin{cases}\text{x}^\text{m}\sin(\frac{1}{\text{x}}), &\text{x}\neq0 \\0 ,& \text{x}=0\end{cases}$
Neirher continuous but not diffierentiable, if $\text{m}\leq0$
→
Integrate the following w. r. t. x:

$\frac{2 \log x+3}{x(3 \log x+2)\left[(\log x)^2+1\right]}$

→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 &4 \\ 3 & 1 & 1 \end{bmatrix}$
→
The equation of the tangent at $(2,3)$ on the curve $y^2=a x^3+b$ is $y=4 x-5$. Find the values of $a$ and $b$.
→