JEE Mathematics/ Lakshmikanta Satapathy/ Questions on Indefinite Integration Part 13, taken from previous Board Papers solved by the method of substitution using standard integrals
JEE Mathematics/ Lakshmikanta Satapathy/ Indefinite Integrals/ Questions on Integration by parts involving trigonometric, inverse trigonometric and logarithmic functions solved with substitution
JEE Mathematics/ Lakshmikanta Satapathy/ Indefinite Integrals/ Questions on Integration by parts involving trigonometric, inverse trigonometric and logarithmic functions solved with substitution
Ejercicios resueltos sobre Transformada de Laplace y su inversa; ecuaciones diferenciales con Laplace y la aplicación de Laplace a un circuito electrónico sencillo RLC.
JEE Mathematics/ Lakshmikanta Satapathy/ Inverse trigonometry QA part 6/ Questions on Inverse trigonometric functions involving tan inverse function solved with the related concepts
Formulae for Sum and Difference of two Tan Inverse functions with proof of the results and Relationship between Sine Inverse , Cos Inverse and tan Inverse functions
JEE Mathematics/ Lakshmikanta Satapathy/ Questions on Indefinite Integration Part 11 solved by the method of substitution with recall of standard results
Ejercicios resueltos sobre Transformada de Laplace y su inversa; ecuaciones diferenciales con Laplace y la aplicación de Laplace a un circuito electrónico sencillo RLC.
JEE Mathematics/ Lakshmikanta Satapathy/ Inverse trigonometry QA part 6/ Questions on Inverse trigonometric functions involving tan inverse function solved with the related concepts
Formulae for Sum and Difference of two Tan Inverse functions with proof of the results and Relationship between Sine Inverse , Cos Inverse and tan Inverse functions
JEE Mathematics/ Lakshmikanta Satapathy/ Questions on Indefinite Integration Part 11 solved by the method of substitution with recall of standard results
JEE Mathematics / Lakshmikanta Satapathy / Probability / Multiplication theorem of probability, for two or more events and independent events and Addition theorem of probability
JEE Mathematics/ Lakshmikanta Satapathy/ Questions on Indefinite Integration Part 14 taken from previous Board papers are solved using integration by parts with alternate methods
Scalars and vectors, Parallel and anti-parallel vectors, equal and negative vectors, collinear, co-planar and co-initial vectors, vector multiplied by scalar
JEE Mathematics / Lakshmikanta Satapathy / Problems on Probability involving a pack of cards, selection of students and colored marbles solved by axiomatic method
JEE Mathematics / Lakshmikanta Satapathy / Questions and Answers on Probability involving Independent events, mutually exclusive events in experiments of drawing colored balls from bags and problem solving probability of students using Addition theorem and Multiplication theorem
JEE Mathematics / Lakshmikanta Satapathy / Questions and Answers on Conditional Probability which includes Problems on two dice and selecting questions from a question bank
JEE Mathematics / Lakshmikanta Satapathy / Problems on determination of Probability involving Union, Intersection and Complement of Events solved by Axiomatic approach
Questions and Answers on Finding the values with sign of the other five Trigonometric Functions from the given value of a Trigonometric function with their representation in a right angled triangle
JEE Physics/ Lakshmikanta Satapathy/ Work Energy and Power/ Force and Potential energy/ Angular momentum and Speed of Particle/ MCQ one or more correct
JEE Physics/ Lakshmikanta Satapathy/ MCQ On Work Energy Power/ Work-Energy theorem/ Work done by Gravity/ Work done by Air resistance/ Change in Kinetic Energy of body
CBSE Physics/ Lakshmikanta Satapathy/ Electromagnetism QA/ Magnetic field due to circular coil at center & on the axis/ Magnetic field due to Straight conductor/ Magnetic Lorentz force
1) Four point charges placed at the corners of a square were given. The total electric potential at the center of the square was calculated to be 4.5 x 10^4 V.
2) The electric field and potential due to a point charge were given. Using these, the distance of the point from the charge and the magnitude of the charge were calculated.
3) An oil drop carrying a charge between the plates of a capacitor was given. The voltage required to balance the drop, given the mass and distance between plates, was calculated to be 9.19 V.
This document discusses the reflection and transmission of waves at the junction of two strings with different linear densities. It provides equations relating the amplitudes of the incident, reflected, and transmitted waves based on the continuity of displacement and slope at the junction. It also discusses sound as a pressure wave and derives an expression for the speed of sound in a fluid from the definition of pressure as a cosine wave. Finally, it defines the loudness of sound in decibels and calculates differences in loudness for different sound intensities.
1) Vibrations in air columns inside closed and open pipes produce standing waves with characteristic frequencies called harmonics or overtones.
2) In closed pipes, only odd harmonics like the fundamental, 1st overtone (3rd harmonic) and 2nd overtone (5th harmonic) are possible. In open pipes, all harmonics including the fundamental, 1st overtone (2nd harmonic) and 2nd overtone (3rd harmonic) are observed.
3) There is an end correction of about 0.3 times the pipe diameter that must be added to the effective pipe length to account for vibrations outside the physical opening.
4) The speed of sound in air can be measured
CBSE Physics/ Lakshmikanta Satapathy/ Wave Motion Theory/ Reflection of waves/ Traveling and stationary waves/ Nodes and anti-nodes/ Stationary waves in strings/ Laws of transverse vibration of stretched strings
CBSE Physics/ Lakshmikanta Satapathy/ Wave theory/ path difference and Phase difference/ Speed of sound in a gas/ Intensity of wave/ Superposition of waves/ Interference of waves
JEE Mathematics/ Lakshmikanta Satapathy/ Definite integrals part 8/ JEE question on definite integral involving integration by parts solved with complete explanation
JEE Physics/ Lakshmikanta Satapathy/ Question on the magnitude and direction of the resultant of two displacement vectors asked by a student solved in the slides
JEE Mathematics/ Lakshmikanta Satapathy/ Quadratic Equation part 2/ Question on properties of the roots of a quadratic equation solved with the related concepts
JEE Mathematics/ Lakshmikanta Satapathy/ Probability QA part 12/ JEE Question on Probability involving the complex cube roots of unity is solved with the related concepts
This document contains two problems from inverse trigonometry. The first problem involves finding the values of x and y given trigonometric expressions involving tan(x) and tan(y). The second problem proves the identity x = -x + pi for x in the range (-pi, pi). Both problems are solved step-by-step using trigonometric identities and properties. The document also provides contact information for the physics help website.
This document discusses the transient current in an LR circuit with two inductors (L1 and L2) and a resistor connected to a 5V battery. It provides the equations for calculating the transient current in an LR circuit. It then calculates that for L1, the ratio of maximum to minimum current (Imax/Imin) is 8. Similarly, for L2 the ratio is 5. The total maximum current drawn from the battery is 40A and the minimum is 5A, giving a ratio of 8.
JEE Physics/ Lakshmikanta Satapathy/ Electromagnetism QA part 7/ Question on doubling the range of an ammeter by shunting solved with the related concepts
JEE Mathematics/ Lakshmikanta Satapathy/ Binomial theorem part 6/ JEE Question on the coefficient of a given power of x in the expansion of 100 factors
2. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
Answer-1
sec 1.I x dx
(1 cos )(1 cos )
.
cos (1 cos )
x x
dx
x x
1 cos
.
cos
x
dx
x
2
sin
.
cos cos
x
dx
x x
1
1 .
cos
dx
x
3. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
cos sin . sin .Put x t x dx dt x dx dt Substitution :
2
2 1 1
4 4
dt dt
I
t t t t
2 2
1 1 1
log
2 2 2
t t C
21
log
2
t t t C
2 2
1 1
2 2
dt
t
2 2
2 2
log
dx
using x x a
x a
2
[ ]
1
log cos cos cos
2
x x x C Ans
4. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
Answer-2
Substitution : sin cos .Put x t x dx dt
2
2sin2 cos
.
6 cos 4sin
x x
I dx
x x
2
(4sin 1)cos
.
sin 4sin 5
x x
dx
x x
2
(4 1)
.
4 5
t
I dt
t t
1 22 2
2 4
2 . 7 ( )
4 5 4 5
t dt
dt I I say
t t t t
2 2
sin2 2sin cos
cos 1 sin
x x x
x x
2
4sin cos cos
.
6 (1 sin ) 4sin
x x x
dx
x x
2 2
4 8 7 2(2 4) 7
. .
4 5 4 5
t t
dt dt
t t t t
5. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
2 1
1 2 2log 4 5 7tan ( 2)I I I t t t C
2 1
2log sin 4sin 5 7tan (sin 2) [ ]Anx x x C s
1 2
2 4
2 .
4 5
t
I dt
t t
2 2 2
7 7
4 5 4 4 1
dt dt
And I
t t t t
2
4 5 (2 4)Put t t u t dt du
2
1 2 2log 2log 4 5
du
I u t t
u
1
2 2
7 7tan ( 2)
( 2) 1
dt
t
t
1
2 2
1
tan
dx x
using
x a a a
6. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
Answer-3
2
2
.
2 1
x
I dx
x x
1 22 2
1 2 2
.
2 2 1 2 1
x dx
dx I I
x x x x
2
1 (2 2) 2
.
2 2 1
x
dx
x x
2
1 2 4
.
2 2 1
x
dx
x x
1 2
1 2 2
.
2 2 1
x
I dx
x x
2
2 1 (2 2).Put x x t x dx dt
2
1 2 1
2
dt
I t x x
t
7. Physics Helpline
L K Satapathy
Indefinite Integrals - 13
2 2 2
2 1 2 1 2
dx dx
And I
x x x x
2 2
2 1 log ( 1) 2 [ ]1x x x x C Ansx
2
2
log ( 1) ( 1) 2x x 2 2
2 2
log
dx
using x x a
x a
2
log ( 1) 2 1x x x
1 2I I I
2
2
( 1) 2
dx
x
8. Physics Helpline
L K Satapathy
For More details:
www.physics-helpline.com
Subscribe our channel:
youtube.com/physics-helpline
Follow us on Facebook and Twitter:
facebook.com/physics-helpline
twitter.com/physics-helpline