
How do you use substitution to integrate respect to x ?
Answer
465.3k+ views
Hint: The given question requires us to integrate a function of x with respect to x. Integration gives us a family of curves. Integrals in maths are used to find many useful quantities such as areas, volumes, displacement, etc. Integral is always found with respect to some variable, which in this case is x.
Complete step by step solution:
The given question requires us to integrate an exponential function in variable x.
So, we have to compute
The power or exponent is linear but not x simply. So, we need to do substitution in order to find the integral of the exponential function.
We can assign a new variable to the power of the exponential function .
Let us assume .
Differentiating both sides of the equation, we get,
Shifting from left side of the equation to right side of the equation to find the value of , we get,
So, the integral can be simplified by substituting the value of as obtained above.
So,
We know that the integral of with respect to x is . So, we get,
Putting the value of t back into the expression, we get,
So, is the integral for the given function with respect to x.
So, the correct answer is “ ”.
Note: The indefinite integrals of certain functions may have more than one answer in different forms. However, all these forms are correct and interchangeable into one another. Indefinite integral gives us the family of curves as we don’t know the exact value of the constant.
Complete step by step solution:
The given question requires us to integrate an exponential function
So, we have to compute
The power or exponent is linear but not x simply. So, we need to do substitution in order to find the integral of the exponential function.
We can assign a new variable to the power of the exponential function
Let us assume
Differentiating both sides of the equation, we get,
Shifting
So, the integral
So,
We know that the integral of
Putting the value of t back into the expression, we get,
So,
So, the correct answer is “
Note: The indefinite integrals of certain functions may have more than one answer in different forms. However, all these forms are correct and interchangeable into one another. Indefinite integral gives us the family of curves as we don’t know the exact value of the constant.
Latest Vedantu courses for you
Grade 11 Science PCM | CBSE | SCHOOL | English
CBSE (2025-26)
School Full course for CBSE students
₹41,848 per year
Recently Updated Pages
Master Class 12 Biology: Engaging Questions & Answers for Success

Class 12 Question and Answer - Your Ultimate Solutions Guide

Master Class 12 Business Studies: Engaging Questions & Answers for Success

Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Social Science: Engaging Questions & Answers for Success

Master Class 12 English: Engaging Questions & Answers for Success

Trending doubts
Which are the Top 10 Largest Countries of the World?

Why is insulin not administered orally to a diabetic class 12 biology CBSE

a Tabulate the differences in the characteristics of class 12 chemistry CBSE

Why is the cell called the structural and functional class 12 biology CBSE

The total number of isomers considering both the structural class 12 chemistry CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
