Sign in to unlock the answer

Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

To integrate the function $\int_{0}^{1} \frac{x^{3}}{1+x^{2}} dx$, we look for a method to simplify the integrand

step 2

We notice that the derivative of $1+x^2$ is $2x$, which is close to the numerator $x^3$. This suggests a substitution

step 3

Let $u = 1+x^2$. Then, $du = 2x dx$

step 4

We need to adjust the numerator to match our substitution. We can write $x^3 dx$ as $\frac{1}{2}x^2 \cdot 2x dx$

step 5

Substituting $u$ and $du$ into the integral, we get $\frac{1}{2}\int \frac{u-1}{u} du$

step 6

Split the integral: $\frac{1}{2}\int 1 - \frac{1}{u} du$

step 7

Integrate term by term: $\frac{1}{2}(u - \ln|u|) + C$

step 8

Substitute back for $u$: $\frac{1}{2}(x^2 - \ln|x^2+1|) + C$

step 9

Evaluate the definite integral from 0 to 1: $\frac{1}{2}[(1^2 - \ln|1^2+1|) - (0^2 - \ln|0^2+1|)]$

step 10

Simplify to get the final answer: $\frac{1}{2}(1 - \ln(2))$

Answer

$\frac{1}{2}(1 - \ln(2)) \approx 0.15343$

Key Concept

Integration by Substitution

Explanation

To integrate a function of the form $\frac{x^n}{1+x^2}$, where $n$ is an odd integer, we can use substitution with $u = 1+x^2$, simplifying the integration process.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question