\begin{tabular}{l} (1 point) Calculate the following antiderivatives: \ \begin{tabular}{l|l} (a) $$ \int 12 t-6 t^{9}-5 d t= $$ \ (b) $$ \int \frac{1}{u^{9 / 4}}+7 \sqrt{u} d u= $$ & $$ \# $$ \ (c) $$ \int \frac{1}{4 x^{3}} d x= $$ & $$ \#+C $$\end{tabular} \ \hline\end{tabular}

Question

UpStudy AI · Accepted Answer

### (a) Evaluate $$ \int (12t - 6t^{9} - 5)\,dt $$

1. **Integrate term-by-term:**

- For $$12t$$:
     $$
     \int 12t\,dt = 12\cdot\frac{t^2}{2} = 6t^2
     $$
     
   - For $$-6t^9$$:
     $$
     \int -6t^9\,dt = -6\cdot\frac{t^{10}}{10} = -\frac{3t^{10}}{5}
     $$
     
   - For $$-5$$:
     $$
     \int -5\,dt = -5t
     $$

2. **Combine the results:**

$$
   \int (12t - 6t^{9} - 5)\,dt = 6t^2 - \frac{3t^{10}}{5} - 5t + C
   $$

---

### (b) Evaluate $$ \int \left( \frac{1}{u^{9/4}} + 7\sqrt{u} \right) du $$

1. **Express the integrand with exponents:**

- Note that $$ \frac{1}{u^{9/4}} = u^{-9/4} $$.
   - Also, $$ \sqrt{u} = u^{1/2} $$.

2. **Integrate each term:**

- For $$ u^{-9/4} $$:
     $$
     \int u^{-9/4}\,du = \frac{u^{-9/4+1}}{-9/4+1} = \frac{u^{-5/4}}{-5/4} = -\frac{4}{5} u^{-5/4}
     $$
     
   - For $$7u^{1/2}$$:
     $$
     \int 7u^{1/2}\,du = 7\cdot \frac{u^{1/2+1}}{1/2+1} = 7\cdot \frac{u^{3/2}}{3/2} = \frac{14}{3} u^{3/2}
     $$

3. **Combine the results:**

$$
   \int \left( \frac{1}{u^{9/4}} + 7\sqrt{u} \right) du = -\frac{4}{5} u^{-5/4} + \frac{14}{3} u^{3/2} + C
   $$

---

### (c) Evaluate $$ \int \frac{1}{4x^{3}}\,dx $$

1. **Factor out the constant:**

$$
   \int \frac{1}{4x^3}\,dx = \frac{1}{4} \int x^{-3}\,dx
   $$

2. **Integrate $$ x^{-3} $$:**

$$
   \int x^{-3}\,dx = \frac{x^{-3+1}}{-3+1} = \frac{x^{-2}}{-2} = -\frac{1}{2} x^{-2}
   $$

3. **Multiply by the constant:**

$$
   \frac{1}{4} \left(-\frac{1}{2}x^{-2}\right) = -\frac{1}{8} x^{-2}
   $$

4. **Final answer:**

$$
   \int \frac{1}{4x^{3}}\,dx = -\frac{1}{8x^2} + C
   $$
(a) $$ \int (12t - 6t^{9} - 5)\,dt = 6t^2 - \frac{3t^{10}}{5} - 5t + C $$ (b) $$ \int \left( \frac{1}{u^{9/4}} + 7\sqrt{u} \right) du = -\frac{4}{5} u^{-5/4} + \frac{14}{3} u^{3/2} + C $$ (c) $$ \int \frac{1}{4x^{3}}\,dx = -\frac{1}{8x^2} + C $$

(1 point) Calculate the following antiderivatives:

(a)

(b)

(c)

Upstudy AI Solution

Answer

Solution

(a) Evaluate

(b) Evaluate

© Evaluate

The Deep Dive

Related Questions

Latest Calculus Questions

(1 point) Calculate the following antiderivatives: (a) (b) (c)