Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
Evaluate the following integrals:
$\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx},\text{ n}\in\text{N},\text{n}\leq2$
✓

Answer

Let $\text{I}=\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx}\ ...(\text{i})$
Then,
$\text{I}=\int\limits^{\text{b}}_{\text{a}}\frac{\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}{\big(\text{a}+\text{b}-\text{x}\big)^\frac{1}{\text{n}}+\big[\text{a}+\text{b}-\big(\text{a}+\text{b}-\text{x}\big)\big]^{\frac{1}{\text{n}}}}\text{ dx}$ $\Bigg[\int\limits^{\text{b}}_{\text{a}}\text{f(x)}\text{dx}=\int\limits^{\text{b}}_{\text{a}}\text{f}(\text{a}+\text{b}-\text{x})\text{dx}\Bigg]$
$=\int\limits^{\text{b}}_{\text{a}}\frac{\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}{\big(\text{a}+\text{b}-\text{x}\big)^\frac{1}{\text{n}}+\text{x}^{\frac{1}{\text{n}}}}\text{ dx}$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{^{\frac{1}{\text{n}}}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx}$
$\Rightarrow2\text{I}=\int\limits^{\text{b}}_{\text{a}}\text{dx}$
$\Rightarrow2\text{I}=\big[\text{x}\big]^{\text{b}}_{\text{a}}=(\text{b}-\text{a})$
$\Rightarrow\text{I}=\frac{\text{b}-\text{a}}{2}$

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 "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

Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.
→
Sketch the graph y = |x + 3|. Evaluate $\int\limits_{-6}^{0}|\text{x}-3|\text{dx} $ . What does this value of the integral represent on the graph.
→
Find: $\int\frac{3\text{x}+5}{\text{x}^2+3\text{x}-18}\text{ dx}.$
→
Consider the binary operation * and o defined by the following tables on set S = {a, b, c, d}.
o
a
b
c
d
a
a
a
a
a
b
a
b
c
d
c
a
c
d
b
d
a
d
b
c
→
A beam is supported at the two ends and is uniformly loaded. The bending moment M at a distance x from one end is given by
$\text{M}=\frac{\text{WL}}{2}\text{x}-\frac{\text{W}}{3}\frac{\text{x}^{3}}{\text{L}^{2}}$
Find the point at which M is maximum in each case.
→
Evaluate the following integrals as limit of sum:
$\int\limits^{\frac{\pi}{2}}_{0}\cos\text{x dx}$
→
A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it. He has a sum of Rs. 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 sq.m of ground per tree and costs Rs. 20 per tree and type B requires 20 sq.m of ground per tree and costs Rs. 25 per tree. When fully grown, type A produces an average of 20kg of fruit which can be sold at a profit of Rs. 2.00 per kg and type B produces an average of 40kg of fruit which can be sold at a profit of Rs. 1.50 per kg. How many of each type should be planted to achieve maximum profit when the trees are fully grown? What is the maximum profit?
→
In a binomial distribution the sum and product of the mean and the variance are $\frac{25}{3}$ and $\frac{50}{3}$ respectively. Find the distribution.
→
Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.
→
Find the angle of intersecting of the following curves:
$\text{x}^2+\text{y}^2=2\text{x}\text{ and }\text{y}^2=8\text{x}$
→