Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 7 Definite Integrals to help strengthen their preparations for exams.

## TS Inter 2nd Year Maths 2B Definite Integrals Important Questions

Very Short Answer Type Questions

Question 1.

Evaluate \(\int_1^2 x^5\) dx

Solution:

\(\int_1^2 x^5 d x=\left[\frac{x^6}{6}\right]_1^2=\frac{2^6}{6}-\frac{1}{6}=\frac{64}{6}-\frac{1}{6}=\frac{63}{6}=\frac{21}{2}\)

Question 2.

Evaluate \(\int_0^\pi \) sinx dx

Solution:

Question 3.

Evaluate \(\int_0^a \frac{d x}{x^2+a^2}\)

Solution:

Question 4.

Evaluate \(\int_1^4 x \sqrt{x^2-1}\) dx

Solution:

Let x^{2} – 1 – t ⇒ 2x dx dt then

Upper limit when x = 4 is t = 15.

Lower Limit when x = 1 is t = 0.

Question 5.

Evaluate \(\int_0^2 \sqrt{4-x^2}\) dx

Solution:

Let x= 2 sin θ = dx – 2cosθ dθ then

Question 6.

Show that \(\int_0^{\frac{\pi}{2}} \sin ^n x d x=\int_0^{\frac{\pi}{2}} \cos ^n x dx\)

Solution:

Question 7.

Evaluate \(\int_0^{\frac{\pi}{2}} \frac{\cos ^{\frac{5}{2}} x}{\sin ^{\frac{5}{2}} x+\cos ^{\frac{5}{2}} x}\) dx

Solution:

Question 8.

Evaluate \(\int_0^{\frac{\pi}{2}} \) x sin x dx

Solution:

Question 9.

Evaluate

Solution:

(iii) \(\int_0^{\frac{\pi}{2}} \sin ^6 x \cos ^4 x dx\)

Solution:

Question 10.

Find \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^2 x \cos ^4 x d x\)

Solution:

Short Answer Type Questions

Question 1.

Find \(\int_0^2\left(x^2+1\right) dx\) as the limit of a sum

Solution:

Question 2.

Evaluate \(\int_0^2 e^x dx\) as the limit of a sum.

Solution:

Question 3.

Lets define f : [0,1]→ R by

f(x) = 1 if x is rational

= 0 if x is irrational

then show that f is nor R Integrable over [0, 1].

Solution:

Let P = (x_{0}, x_{1},…., x_{n}] be a partition of [0, 1].

Since between any two real numbers there exists rational and irrational numbers and

let t_{i}, s_{i} ∈ [X_{i} -i x_{j}] be the rational and irrational numbers.

Question 4.

Evalute \(\int_0^{16} \frac{x^{\frac{1}{4}}}{1+x^{\frac{1}{2}}}\) dx

Solution:

Let x = t^{4} then dx – 4t^{3} dt

Upper limit when x = 16 is t = 2.

and Lower limit when x = 0 is t = 0.

Question 5.

Evaluate \(\int_{-\frac{\pi}{2}}^\pi \sin\) |x| dx

Solution:

We have sin |x| = sin(-x) if x < 0

= sinx if x ≥ 0

Question 6.

Evaluate by using the method of finding definite integral as the limit of a sum.

\(\lim _{n \rightarrow \infty} \sum_{i=1}^n \frac{1}{n}\left(\frac{n-1}{n+1}\right)\)

Solution:

Question 7.

Evaluate \(\lim _{n \rightarrow \infty} \frac{2^k+4^k+6^k+\ldots+(2 n)^k}{n^{k+1}}\) using the method of finding definite integral as the limit of a sum.

Solution:

Question 8.

Evaluate \(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \ldots\left(1+\frac{n}{n}\right)\right]^{\frac{1}{n}}\)

Solution:

Question 9.

Obtain Reduction formula for \(\int_0^{\frac{\pi}{2}} \sin ^n x d x\) and hence find

Solution:

Question 10.

Evaluate \(\int_0^a \sqrt{a^2-x^2} dx\)

Solution:

Let x = a sinθ then dx = a cosθ dθ

Upper limit when x = a is θ = \(\frac{\pi}{2}\)

and Lower limit when x = 0 is θ = 0

Question 11.

Find \(\int_{-a}^a x^2\left(a^2-x^2\right)^{3 / 2} dx\)

Solution:

Since f(x) = x^{2} (a^{2} – x^{2})^{3/2 }is an even function and f(- x) = f(x) we have

\(\int_{-a}^a x^2\left(a^2-x^2\right)^{3 / 2} d x=2 \int_0^a x^2\left(a^2-x^2\right)^{3 / 2} d x\)

Let x = a sin θ then dx = a cos θ dθ

∴ Upper limit when x = a is θ = \(\frac{\pi}{2}\)

Lower limit when x = 0 is θ = 0

Question 12.

Find \(\int_0^1 x^{3 / 2} \sqrt{1-x} dx\)

Solution:

Let x = sin^{2}θ then dx = 2 sinθ cosθ dθ

Upper limit when x = 1 is θ = \(\frac{\pi}{2}\)

Lower Limit when x = θ is θ = 0.

Question 13.

Find the area under the curve f(x) = sin x in (0, 2π).

Solution:

Consider the graph of the function f(x) = sinx in [0, 2π];

we have sin x ≥ 0 ∀ x ∈ [0,π] and sin x≤0∀x∈[π,2π].

Question 14.

Find the area under the curve f(x) = cos x in [0, 2π].

Solution:

Question 15.

Find the bounded by the y = x^{2} parabola the X- axis and the lines x = – 1, x = 2.

Solution:

Question 16.

Find the area cut off between the line y = 0 and the parabola y = x^{2}– 4x + 3.

Solution:

The point of intersection of y – 0 and y = x^{2} – 4x + 3 is given by x^{2} – 4x + 3 = 0

= (x – 3)(x-1) = 0 = x = 1 or 3

y=x^{2}– 4x + 3 ⇒ y+1 = x^{2}– 4x + 4 (x-2)^{2}

Hence the equation represents a parabola

with vertex (2, -1) lies in IV quadrant.

Question 17.

Find the area bounded by the curves y = sin x and y = cos x between any two consecutive points of intersection.

Solution:

The given curves y = sin x and y = cosx and

tan x = 1 ⇒ x = \(\frac{\pi}{4}\)

∴ x = \(\frac{\pi}{4}\) and x = \(\frac{5 \pi}{4}\) are the two consecutive points of intersection.

Taking f(x) = sin x and g(x) cos x over \(\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]\) we have

f(x)> g(x) ∀ x ∈\(\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]\).

Hence the area bounded by y = sin x, y = cos x and the two points of intersection is

Question 18.

Find the area of one of the curvilinear rectangles bounded by y = sin x, y cos x and X-axis.

Solution:

Question 19.

Find the area of the right angled triangle with base b and altitude ‘h’ using the fundamental theorem of integral calculus.

Solution:

Question 20.

Find the area bounded between the curves y^{2} – 1 = 2x and x = 0.

Solution:

The given curves are

y^{2}-1-2x-2(x-0) ……………. (1)

= (y-0)^{2 }2(x)+1=2 \(\left[\mathrm{x}+\frac{1}{2}\right]\)

(1) represents parabola with vertex \(\left(-\frac{1}{2}, 0\right)\)

Solving (1) and x = 0 we get

y^{2} -1 = 0 ⇒ y = ±1

∴ The points of intersection are (0, 1), (0, -1).

The parabola meets the X- axis and y = 1 and y = – 1 and the curve is symmetric with respect to X – axis

Question 21.

Find the area enclosed by the curves y = 3x and y = 6x-x^{2}.

Solution:

Given curves are y^{3}x and y=6x – x^{2
}Solving 6x – x^{2} = 3x = 3x – x^{2} = 0

= x(3- x)=0 =x=0 or x=3

Taking f(x) = 3x and g(x) = 6x – x^{2}

then g(x) ≥ 1(x) in [0, 3] and area enclosed between the line y = 3x and the parabola y = 6x-x^{2} is

Long Answer Type Questions

Question 1.

Show that \(\int_0^{\frac{\pi}{2}} \frac{x}{\sin x+\cos x}\) dx =\(\frac{\pi}{2 \sqrt{2}} \log (\sqrt{2}+1)\)

Solution:

Question 2.

Evaluate \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\) dx

Solution:

Question 3.

Evaluate \(\int_{-a}^a\left(x^2+\sqrt{a^2-x^2}\right) dx\)

Solution:

Question 4.

Evaluate \(\int_0^\pi \frac{x \sin x}{1+\sin x}\) dx

Solution:

Question 5.

Find \(\int_0^\pi \mathbf{x}\) sin^{7} x cos ^{6} x dx.

Solution:

Question 6.

Find the area enclosed between y=x^{2}-5x and y=4-2x.

Solution:

The graphs of curves are shown below.

Question 7.

Find the area bounded between the curves y = x^{2}, y = \(\sqrt{\mathbf{x}} \)

Solution:

Question 8.

Find the area bounded between the curves y^{2}=4ax, x^{2}= 4by(a>0,b>0).

Solution: