Calculus Examples

$\int_{0}^{t} x \sqrt{x - t} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sqrt{x - t}$ .

$x (\frac{2}{3} {(x - t)}^{\frac{3}{2}})]_{0}^{t} - \int_{0}^{t} \frac{2}{3} {(x - t)}^{\frac{3}{2}} d x$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Combine $\frac{2}{3}$ and ${(x - t)}^{\frac{3}{2}}$ .

$x \frac{2 {(x - t)}^{\frac{3}{2}}}{3}]_{0}^{t} - \int_{0}^{t} \frac{2}{3} {(x - t)}^{\frac{3}{2}} d x$

Step 2.2

Combine $x$ and $\frac{2 {(x - t)}^{\frac{3}{2}}}{3}$ .

$\frac{x (2 {(x - t)}^{\frac{3}{2}})}{3}]_{0}^{t} - \int_{0}^{t} \frac{2}{3} {(x - t)}^{\frac{3}{2}} d x$

Step 3

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

$\frac{x (2 {(x - t)}^{\frac{3}{2}})}{3}]_{0}^{t} - (\frac{2}{3} \int_{0}^{t} {(x - t)}^{\frac{3}{2}} d x)$

Step 4

Let $u = x - t$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = x - t$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $x - t$ .

$\frac{d}{d x} [x - t]$

Step 4.1.2

By the Sum Rule, the derivative of $x - t$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- t]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- t]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- t]$

Step 4.1.4

Since $- t$ is constant with respect to $x$ , the derivative of $- t$ with respect to $x$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

Substitute the lower limit in for $x$ in $u = x - t$ .

$u_{lower} = 0 - t$

Step 4.3

Subtract $t$ from $0$ .

$u_{lower} = - t$

Step 4.4

Substitute the upper limit in for $x$ in $u = x - t$ .

$u_{upper} = t - t$

Step 4.5

Subtract $t$ from $t$ .

$u_{upper} = 0$

Step 4.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - t$

$u_{upper} = 0$

Step 4.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{x (2 {(x - t)}^{\frac{3}{2}})}{3}]_{0}^{t} - \frac{2}{3} \int_{- t}^{0} u^{\frac{3}{2}} d u$

Step 5

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

$\frac{x (2 {(x - t)}^{\frac{3}{2}})}{3}]_{0}^{t} - \frac{2}{3} \frac{2}{5} u^{\frac{5}{2}}]_{- t}^{0}$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $\frac{x (2 {(x - t)}^{\frac{3}{2}})}{3}$ at $t$ and at $0$ .

$(\frac{t (2 {(t - t)}^{\frac{3}{2}})}{3}) - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} \frac{2}{5} u^{\frac{5}{2}}]_{- t}^{0}$

Step 6.2

Evaluate $\frac{2}{5} u^{\frac{5}{2}}$ at $0$ and at $- t$ .

$(\frac{t (2 {(t - t)}^{\frac{3}{2}})}{3}) - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Subtract $t$ from $t$ .

$\frac{t (2 \cdot 0^{\frac{3}{2}})}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.2

Rewrite $0$ as $0^{2}$ .

$\frac{t (2 \cdot {(0^{2})}^{\frac{3}{2}})}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{t (2 \cdot 0^{2 (\frac{3}{2})})}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.4.1

Cancel the common factor.

$\frac{t (2 \cdot 0^{2 (\frac{3}{2})})}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.4.2

Rewrite the expression.

$\frac{t (2 \cdot 0^{3})}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.5

Raising $0$ to any positive power yields $0$ .

$\frac{t (2 \cdot 0)}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.6

Multiply $2$ by $0$ .

$\frac{t \cdot 0}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.7

Multiply $t$ by $0$ .

$\frac{0}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.8

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 6.3.8.1

Factor $3$ out of $0$ .

$\frac{3 (0)}{3} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.8.2

Cancel the common factors.

Tap for more steps...

Step 6.3.8.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 0}{3 \cdot 1} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.8.2.2

Cancel the common factor.

$\frac{3 \cdot 0}{3 \cdot 1} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.8.2.3

Rewrite the expression.

$\frac{0}{1} - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.8.2.4

Divide $0$ by $1$ .

$0 - \frac{0 (2 {(0 - t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.9

Subtract $t$ from $0$ .

$0 - \frac{0 (2 {(- t)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.10

Factor $- 1$ out of $- t$ .

$0 - \frac{0 (2 {(- (t))}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.11

Apply the product rule to $- (t)$ .

$0 - \frac{0 (2 ({(- 1)}^{\frac{3}{2}} t^{\frac{3}{2}}))}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.12

Multiply $2$ by $0$ .

$0 - \frac{0 ({(- 1)}^{\frac{3}{2}} t^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.13

Multiply ${(- 1)}^{\frac{3}{2}}$ by $0$ .

$0 - \frac{0 t^{\frac{3}{2}}}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.14

Multiply $0$ by $t^{\frac{3}{2}}$ .

$0 - \frac{0}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.15

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 6.3.15.1

Factor $3$ out of $0$ .

$0 - \frac{3 (0)}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.15.2

Cancel the common factors.

Tap for more steps...

Step 6.3.15.2.1

Factor $3$ out of $3$ .

$0 - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.15.2.2

Cancel the common factor.

$0 - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.15.2.3

Rewrite the expression.

$0 - \frac{0}{1} - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.15.2.4

Divide $0$ by $1$ .

$0 - 0 - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.16

Multiply $- 1$ by $0$ .

$0 + 0 - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.17

Add $0$ and $0$ .

$0 - \frac{2}{3} ((\frac{2}{5} \cdot 0^{\frac{5}{2}}) - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.18

Rewrite $0$ as $0^{2}$ .

$0 - \frac{2}{3} (\frac{2}{5} \cdot {(0^{2})}^{\frac{5}{2}} - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.19

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$0 - \frac{2}{3} (\frac{2}{5} \cdot 0^{2 (\frac{5}{2})} - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.20

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.20.1

Cancel the common factor.

$0 - \frac{2}{3} (\frac{2}{5} \cdot 0^{2 (\frac{5}{2})} - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.20.2

Rewrite the expression.

$0 - \frac{2}{3} (\frac{2}{5} \cdot 0^{5} - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.21

Raising $0$ to any positive power yields $0$ .

$0 - \frac{2}{3} (\frac{2}{5} \cdot 0 - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.22

Multiply $\frac{2}{5}$ by $0$ .

$0 - \frac{2}{3} (0 - \frac{2}{5} {(- t)}^{\frac{5}{2}})$

Step 6.3.23

Factor $- 1$ out of $- t$ .

$0 - \frac{2}{3} (0 - \frac{2}{5} {(- (t))}^{\frac{5}{2}})$

Step 6.3.24

Apply the product rule to $- (t)$ .

$0 - \frac{2}{3} (0 - \frac{2}{5} ({(- 1)}^{\frac{5}{2}} t^{\frac{5}{2}}))$

Step 6.3.25

Combine ${(- 1)}^{\frac{5}{2}}$ and $\frac{2}{5}$ .

$0 - \frac{2}{3} (0 - \frac{{(- 1)}^{\frac{5}{2}} \cdot 2}{5} t^{\frac{5}{2}})$

Step 6.3.26

Combine $t^{\frac{5}{2}}$ and $\frac{{(- 1)}^{\frac{5}{2}} \cdot 2}{5}$ .

$0 - \frac{2}{3} (0 - \frac{t^{\frac{5}{2}} ({(- 1)}^{\frac{5}{2}} \cdot 2)}{5})$

Step 6.3.27

Move $2$ to the left of ${(- 1)}^{\frac{5}{2}}$ .

$0 - \frac{2}{3} (0 - \frac{t^{\frac{5}{2}} (2 \cdot {(- 1)}^{\frac{5}{2}})}{5})$

Step 6.3.28

Move $2$ to the left of $t^{\frac{5}{2}}$ .

$0 - \frac{2}{3} (0 - \frac{2 \cdot t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}}}{5})$

Step 6.3.29

Subtract $\frac{2 t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}}}{5}$ from $0$ .

$0 - \frac{2}{3} (- \frac{2 t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}}}{5})$

Step 6.3.30

Multiply $- 1$ by $- 1$ .

$0 + 1 (\frac{2}{3}) \frac{2 t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}}}{5}$

Step 6.3.31

Multiply $\frac{2}{3}$ by $1$ .

$0 + \frac{2}{3} \cdot \frac{2 t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}}}{5}$

Step 6.3.32

Multiply $\frac{2}{3}$ by $\frac{2 t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}}}{5}$ .

$0 + \frac{2 (2 t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}})}{3 \cdot 5}$

Step 6.3.33

Multiply $2$ by $2$ .

$0 + \frac{4 (t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}})}{3 \cdot 5}$

Step 6.3.34

Multiply $3$ by $5$ .

$0 + \frac{4 (t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}})}{15}$

Step 6.3.35

Add $0$ and $\frac{4 t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}}}{15}$ .

$\frac{4 t^{\frac{5}{2}} {(- 1)}^{\frac{5}{2}}}{15}$

Step 7

Reorder terms.

$\frac{4 {(- 1)}^{\frac{5}{2}}}{15} t^{\frac{5}{2}}$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

Rewrite ${(- 1)}^{\frac{5}{2}}$ as $\sqrt{{(- 1)}^{5}}$ .

$\frac{4 \sqrt{{(- 1)}^{5}}}{15} t^{\frac{5}{2}}$

Step 8.1.2

Raise $- 1$ to the power of $5$ .

$\frac{4 \sqrt{- 1}}{15} t^{\frac{5}{2}}$

Step 8.1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{4 i}{15} t^{\frac{5}{2}}$

Step 8.2

Combine $\frac{4 i}{15}$ and $t^{\frac{5}{2}}$ .

$\frac{4 i t^{\frac{5}{2}}}{15}$