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
$u=\sqrt{1-t}$
Intermediate steps
2
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
Substituting $u$, $dt$ and $t$ in the integral and simplify
$\int-2u^2\left(-u^{2}+1\right)^2du$
Intermediate steps
6
Rewrite the integrand $-2u^2\left(-u^{2}+1\right)^2$ in expanded form
$\int\left(-2u^{6}+4u^{4}-2u^2\right)du$
7
Expand the integral $\int\left(-2u^{6}+4u^{4}-2u^2\right)du$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
$\int-2u^{6}du+\int4u^{4}du+\int-2u^2du$
Intermediate steps
8
The integral $\int-2u^{6}du$ results in: $-\frac{2}{7}\sqrt{\left(1-t\right)^{7}}$
$-\frac{2}{7}\sqrt{\left(1-t\right)^{7}}$
Intermediate steps
9
The integral $\int4u^{4}du$ results in: $\frac{4}{5}\sqrt{\left(1-t\right)^{5}}$
$\frac{4}{5}\sqrt{\left(1-t\right)^{5}}$
Intermediate steps
10
The integral $\int-2u^2du$ results in: $-\frac{2}{3}\sqrt{\left(1-t\right)^{3}}$
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more