Scribd
Upload
56 views2 pages

Problem Set - 6: MATHEMATICS-II (MA10002)

1. Evaluate an integral involving logarithms and trigonometric functions using differentiation under the integral sign. 2. Use differentiation under the integral sign to prove several integral identities involving trigonometric and exponential functions. 3. For a given differential equation relating dx/dy and an integral involving x, (a) find dx/dy and (b) evaluate a definite integral in terms of x. 4. Show that a definite integral involving logarithms and trigonometric functions equals an inverse sine expression, despite a point of discontinuity in the integrand.

Uploaded by

Maulindu
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
56 views2 pages

Problem Set - 6: MATHEMATICS-II (MA10002)

1. Evaluate an integral involving logarithms and trigonometric functions using differentiation under the integral sign. 2. Use differentiation under the integral sign to prove several integral identities involving trigonometric and exponential functions. 3. For a given differential equation relating dx/dy and an integral involving x, (a) find dx/dy and (b) evaluate a definite integral in terms of x. 4. Show that a definite integral involving logarithms and trigonometric functions equals an inverse sine expression, despite a point of discontinuity in the integrand.

Uploaded by

Maulindu
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

Problem Set - 6 Spring 2019

MATHEMATICS-II(MA10002)
Rθ
1. Evaluate 0 log(1 + tan θ tan x) dx by using differentiation under the
integral sign, where −π
2
< θ < π2 .

2. Using differentiation under the integral sign prove the following :-


R ∞ −1 (ax)
(a) 0 tan x(1+x2 )
dx = π2 log(a + 1), a ≥ 0, a 6= 1.
R ∞ −x2 √ R ∞ −x2
π
(b) Using
√
the result 0
e dx = 2
, show that 0
e cos(2αx) dx =
π −α2
2
e .
R a log(1+ax)
(c) 0 1+x2 dx = 12 log(1 + a2 ) tan−1 (a), a ≥ 0 and hence show
R1
that 0 log(1+x)
1+x2
dx = π8 log 2
dy
R 2y 2
3. (a) Find dx if √x e−xt dt = −2, x > 0.
Rx
(b) Evaluate f (x), when f 0 (x) = 3 − 1 fu(u) 2 du, given that f (1) = 1.

Rπ
4. Show that 0 log(1+a cos x)
cos x
dx = π sin−1 (a), |a| < 1, although x = π2 is a
point of discontinuity of the integrand.

5. For any real numbers x and t, let


( 3 xt
(x2 +t2 )2
, if x 6= 0, t 6= 0
f (x, t) =
0, if x = 0, t = 0

and Z 1
F (t) = f (x, t) dx.
0
d
R1 R1 ∂
Is dt 0
f (x, t) dx = 0 ∂t
f (x, t) dx? Give the justification.
R π2
6. (a) Show that π
−α sin θ cos−1 (cos α csc θ) dθ = π
2
(1 − cos α), 0 <
2
α < π/2.
R1 xa −xb
(b) Show that 0 log x
dx = log( a+1
b+1
).
(c) Using Leibnitz’s rule, evaluate
Z cos α
d
log(x + α) dx, when sin α + α > 0, cos α + α > 0
dα sin α
Problem Set - 6 Spring 2019
MATHEMATICS-II(MA10002)

Rπ 1
7. Find the value of 0 a+b cos x
dx when a > 0, |b| < a and deduce that
Z π
1 πa
2
dx = 2 .
0 (a + b cos x) (a − b2 )3/2

8. Evaluate the following integral over the region R:


RR
(a) ydxdy, where R is the region bounded by the parabolas y 2 = 4x
R
and x2 = 4y.
RR 2
(b) x dxdy, where R is the region in the first quadrant bounded
R
by the hyperbola xy = 16 and the lines y = x, y = 0, x = 8.
RR x2
(c) e dxdy, where R is the region given by 2y ≤ x ≤ 2, 0 ≤ y ≤ 1.
R
RR
(d) xydxdy, where R is the domain bounded by the x−axis, ordi-
R
nate x = 2a and the curve x2 = 4ay.

9. Evaluate the following integrals by changing the order of integration:-


RaRa x
(a) 0 y x2 +y 2 dx dy.

R π R π sin y
(b) 02 y=x 2
y
dy dx.
R∞Rx x2
(c) 0 0 xe− y dy dx.
R 3 R √4−y
(d) 0 1 (x + y) dx dy.
√
R 1 R 1−x2 2
(e) 0 0 y dy dx.

*********************************************************************

You might also like