∫(e^(x^2+4 lnx)-x^3 e^x^2)/(x-1) d x is equal to (a) ((e^3...

Key Concepts

Exponential and Logarithmic Properties

Understanding how exponents and logarithms interact is crucial. This includes the property that an exponential expression with a sum in the exponent can be split into a product of exponentials, and how logarithms can be used to transform multiplicative factors into additive ones, thus aiding in simplifying complex expressions before integration.

Algebraic Simplification

Before attempting integration, simplifying the integrand through algebraic manipulation is often essential. This means factoring common terms, canceling out factors when possible, and rewriting expressions in a form that is more conducive to further techniques, which can significantly reduce the complexity of the integration problem.

Substitution Method (u?substitution)

The substitution method is a powerful technique in integration where a part of the integrand is replaced with a new variable. This is especially useful when the derivative of the inner function appears elsewhere in the integrand. It helps in converting the integral into a standard form that is easier to evaluate.

Integration Techniques

Various techniques such as direct integration after simplification, integration by parts, or recognizing the derivative of a known function can be applied once the integral has been simplified. A systematic approach often involves identifying patterns or standard forms in the integrand that suggest a particular integration method.

Transcript

00:01 First, let's simplify this given integral before we integrate it.

00:05 Consider this term e raised to the power of x squared plus 4 times of lnx.

00:13 Let me write down here.

00:17 This can be written as e raised to the power of x squared times of e raised to the power of 4 times of natural logarithm of x.

00:26 And e raised to the power of x squared, this 4 will get raised to the power of x.

00:30 So therefore this becomes e raised to the power of natural logarithm of x power 4 notice that this e and natural logarithm will get cancelled because they are inverse to each other so therefore we get e raised to the power of x squared times of x power 4 and i'm going to replace this inside this integral so therefore the given integral can be written as e raised to the power of x squared times of x power 4 minus of the second term which is x cube times of e rise to the power of x squared over x minus 1 dx and in the next step i'm going to factor this term that is x cube e raise to the power of x squared from both these terms and when i do that i'll be getting x cube e raise to the power of x squared so when i factor this term from the first term i'll be getting only x and from the second term i just get one so i put it as negative 1 over x minus 1 d x what is that this x minus 1 and this x minus 1 in the numerator as plus the denominator the denominator they get cancelled and so we are left with just the integration of x cube e raised to the power of x squared dx to integrate this integral i'm going to put x squared equals z so let me do that x squared equal to and then i differentiate both sides so i get 2x times of dx equal to daz so when i apply this substitution into this given integral this become x cube i can write down this as x squared times of x times of e raised to the power of x squared d x observe that i can solve for x dx after i made the substitution x dx is basically dz by so therefore, we can observe that we have this x times dx, which can be replaced as dz by 2.

02:43 So i put dz by 2 for these two.

02:46 And then x squared, i'm going to replace it as a z.

02:51 So therefore, this becomes a z.

02:54 And for e raise to the power of x squared, that will become e raise to the power of z.

03:00 So this is what we get after we made the substitution.

03:05 Let's rewrite this as i can factor out the 1 by 2 from the integral and we just have to integrate z -e -p -z -d -z.

03:16 Now to integrate this we can apply the integration by parts method, which is shortly written as i -b -p.

03:25 In this we have to follow the rule which is i -l -a -t -e.

03:32 This rule is used to choose the value of u.

03:36 This i stands for integration, logarithmic, algebra, trigonometric, and exponential.

03:43 So whichever comes first in this order, we have to choose that as you.

03:49 Okay, so now we look at our expression...

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