Final Answer
Step-by-step Solution
Specify the solving method
We can solve the integral $\int t^2\sqrt{1-t}dt$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
We can solve the integral $\int\sqrt{1-t}dt$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $1-t$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dt$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dt$ in the previous equation
Substituting $u$ and $dt$ in the integral and simplify
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $\frac{1}{2}$
Replace $u$ with the value that we assigned to it in the beginning: $1-t$
Now replace the values of $u$, $du$ and $v$ in the last formula
We can solve the integral $\int t\sqrt{\left(1-t\right)^{3}}dt$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\sqrt{\left(1-t\right)^{3}}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dt$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dt$ in the previous equation
Rewriting $t$ in terms of $u$
Substituting $u$, $dt$ and $t$ in the integral and simplify
Multiply $\frac{4}{3}$ times $-\frac{2}{3}$
The integral $-\frac{8}{9}\int\sqrt[3]{u^{2}}\left(-\sqrt[3]{u^{2}}+1\right)du$ results in: $\frac{8}{21}\sqrt{\left(1-t\right)^{7}}-\frac{8}{15}\sqrt{\left(1-t\right)^{5}}$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$