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
Differentiate both sides of the equation $u=\sqrt{1-t}$
Find the derivative
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of the constant function ($1$) is equal to zero
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
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
Removing the variable's exponent raising both sides of the equation to the power of $2$
Divide $1$ by $\frac{1}{2}$
Simplify $\left(\sqrt{1-t}\right)^{2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $\frac{1}{2}$ and $n$ equals $2$
Multiply $\frac{1}{2}$ times $2$
Multiply $\frac{1}{2}$ times $2$
Divide $1$ by $\frac{1}{2}$
We need to isolate the dependent variable $t$, we can do that by simultaneously subtracting $1$ from both sides of the equation
Multiply both sides of the equation by $-1$
Solve the product $-\left(u^{2}-1\right)$
Multiply $-1$ times $-1$
Rewriting $t$ in terms of $u$
Rewriting $t$ in terms of $-u^{2}+1$
Solve the product $-\left(-u^{2}+1\right)$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Subtract the values $1$ and $-1$
Cancel exponents $2$ and $\frac{1}{2}$
Divide fractions $\frac{\left(-u^{2}+1\right)^2u}{\frac{-1}{2u}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Multiply $2$ times $-1$
Substituting $u$, $dt$ and $t$ in the integral and simplify
Expand $\left(-u^{2}+1\right)^2$
Multiply the single term $-2u^2$ by each term of the polynomial $\left(u^{4}-2u^{2}+1\right)$
When multiplying exponents with same base we can add the exponents
When multiplying two powers that have the same base ($u^{2}$), you can add the exponents
Simplify $\left(u^{2}\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2$
Simplify $\left(u^{2}\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2$
Multiply $2$ times $2$
Multiply $2$ times $2$
Rewrite the integrand $-2u^2\left(-u^{2}+1\right)^2$ in expanded form
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
The integral of a function times a constant ($-2$) 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 $6$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{1-t}$
The integral $\int-2u^{6}du$ results in: $-\frac{2}{7}\sqrt{\left(1-t\right)^{7}}$
The integral of a function times a constant ($4$) 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 $4$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{1-t}$
The integral $\int4u^{4}du$ results in: $\frac{4}{5}\sqrt{\left(1-t\right)^{5}}$
The integral of a function times a constant ($-2$) 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 $2$
Replace $u$ with the value that we assigned to it in the beginning: $\sqrt{1-t}$
The integral $\int-2u^2du$ 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 $C$