Question:
Evaluate the following integrals:
$\int 2 x^{3} e^{x^{2}} d x$
Solution:
Let $\mathrm{I}=\int 2 \mathrm{x}^{3} \mathrm{e}^{\mathrm{x}^{2}} \mathrm{dx}$
Put $x^{2}=t$
$2 x d x=d t$
$I=\int t e^{t} d t$
Using integration by parts,
$=\mathrm{t} \int \mathrm{e}^{\mathrm{t}} \mathrm{dt}-\int \frac{\mathrm{d}}{\mathrm{dt}} \mathrm{t} \int \mathrm{e}^{\mathrm{t}} \mathrm{dt}$
We have, $\int e^{x} d x=e^{x}$
$=t e^{t}-e^{t}+c$
$=e^{t}(t-1)+c$
Substitute value for $t$,
$I=e^{x^{2}}\left(x^{2}-1\right)+c$
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- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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