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 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 $\sqrt{1-t}$ 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
Rewriting $t$ in terms of $u$
Substituting $u$, $dt$ and $t$ in the integral and simplify
We can solve the integral $\int-2u^2\left(-u^{2}+1\right)^2du$ by applying integration method of trigonometric substitution using the substitution
Now, in order to rewrite $d\theta$ 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
Substituting in the original integral, we get
We can solve the integral $\int-2\sin\left(\theta \right)^2\left(-\sin\left(\theta \right)^{2}+1\right)^2\cos\left(\theta \right)d\theta$ 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 $v$), which when substituted makes the integral easier. We see that $\sin\left(\theta \right)$ it's a good candidate for substitution. Let's define a variable $v$ and assign it to the choosen part
Now, in order to rewrite $d\theta$ in terms of $dv$, we need to find the derivative of $v$. We need to calculate $dv$, we can do that by deriving the equation above
Isolate $d\theta$ in the previous equation
Substituting $v$ and $d\theta$ in the integral and simplify
Rewrite the integrand $-2v^2\left(-v^{2}+1\right)^2$ in expanded form
Expand the integral $\int\left(-2v^{6}+4v^{4}-2v^2\right)dv$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
The integral $\int-2v^{6}dv$ results in: $-\frac{2}{7}\sqrt{\left(1-t\right)^{7}}$
The integral $\int4v^{4}dv$ results in: $\frac{4}{5}\sqrt{\left(1-t\right)^{5}}$
The integral $\int-2v^2dv$ results in: $-\frac{2}{3}\sqrt{\left(1-t\right)^{3}}$
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 $C$