Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals3 Marks
Question
Evaluate the following integrals:$\int\limits^\text{b}_{\text{a}}\frac{\text{f(x)}}{\text{f(x)}+\text{f}(\text{a}+\text{b}-\text{x})}\text{ dx}$
✓

Answer

Let $\text{I}=\int\limits^\text{b}_{\text{a}}\frac{\text{f(x)}}{\text{f(x)}+\text{f}(\text{a}+\text{b}-\text{x})}\text{ dx}\ ....(\text{i})$$=\int\limits^\text{b}_{\text{a}}\frac{\text{f}(\text{a}+\text{b}-\text{x})}{\text{f}(\text{a}+\text{b}-\text{x})+\text{f}(\text{a}+\text{b}-\text{a}-\text{b}+\text{x})}\text{ dx}$
$=\int\limits^\text{b}_{\text{a}}\frac{\text{f}(\text{a}+\text{b}-\text{x})}{\text{f}(\text{a}+\text{b}-\text{x})+\text{f(x)}}\text{ dx}$
$\therefore\ \text{I}=\int\limits^\text{b}_{\text{a}}\frac{\text{f}(\text{a}+\text{b}-\text{x})}{\text{f(x)}+\text{f}(\text{a}+\text{b}-\text{x})}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii) we get$2\text{I}=\int\limits^\text{b}_{\text{a}}\bigg[\frac{\text{f(x)}}{\text{f(x)}+\text{f}(\text{a}+\text{b}-\text{x})}+\frac{\text{f}(\text{a}+\text{b}-\text{x})}{\text{f(x)}+\text{f}(\text{a}+\text{b}-\text{x})}\bigg]\text{dx}$
$=\int\limits^\text{b}_{\text{a}}\frac{\text{f(x)}+\text{f}(\text{a}+\text{b}-\text{x})}{\text{f(x)}+\text{f}(\text{a}+\text{b}-\text{x})}\text{ dx}$
$=\big[\text{x}\big]^{\text{b}}_\text{a}$
$=\text{b}-\text{a}$
Hence, $\text{I}=\frac{\text{b}-\text{a}}{2}$

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.(3 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

Evaluate : $\int e^{\tan ^{-1} x} \cdot\left(\frac{1+x+x^2}{1+x^2}\right) \cdot d x$
→
Find the inverse of each of the following matrices $($if they exist$) : \left[\begin{array}{lll}2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2\end{array}\right]$
→
If $x =\frac{t+1}{t-1}, y =\frac{1-t}{t+1}$, then show that $y 2-\frac{d y}{d x}=0$.
→
Evaluate: $\int_0^{ a } \frac{1}{x+\sqrt{ a ^2-x^2}} d x$
→
If $\text{A}=\begin{bmatrix}1&2\\0&3 \end{bmatrix}$ is written as B + C, where B is a symmetric matrix and C is a skew- symmetric matrix, then B is equal to.
→
Each of the total five questions in a multiple choice examination has four choices, only one of which is correct. A student is attempting to guess the answer. The random variable $x$ is the number of questions answered correctly. What is the probability that the student will give atleast one correct answer ?
→
For any two vectors $\vec{\text{a}}$ and $\vec{\text{b}},$ prove that $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2=\begin{vmatrix}\vec{\text{a}}.\vec{\text{a}}&\vec{\text{a}}.\vec{\text{b}}\\\vec{\text{b}}.\vec{\text{a}}&\vec{\text{b}} .\vec{\text{b}}\end{vmatrix}.$
→
Evaluate the following :

$\int \sqrt{\frac{9+x}{9-x}} \cdot d x$

→
Find k ifThe slope of one of the lines given by $3 x^2+4 x y+k y^2=0$ is three times the other.
→
It is known that $10\%$ of certain articles manufactured are defective. What is the probability that in a random sample of $12$ such articles, $9$ are defective?
→