Q. 15 - Calculus

Question

Find three integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification.

Step-by-Step Solution

Answer

The three integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification are $\int \frac{\sqrt{x}+1}{x}\mathrm{d}x,\int x^4{(x^3+1)}^{2}dx \mathrm{and} \int \frac{2-e^x}{e^x}\mathrm{d}x$.

Step 1. Given Information

Find three integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification.

Step 2. The first integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification.

$\int \frac{\sqrt{x}+1}{x}\mathrm{d}x$

After algebraic simplification

$\int \frac{\sqrt{x}+1}{x}\mathrm{d}x=\int \left(\frac{\sqrt{x}}{x}+\frac{1}{x}\right)\mathrm{d}x=\int \left(\frac{1}{\sqrt{x}}+\frac{1}{x}\right)\mathrm{d}x$

Now we can anti-differentiate immediately.

Step 3. The second integrals in Exercises 21–70 that we can anti differentiate immediately after algebraic simplification.

$\int x^4{(x^3+1)}^{2}dx$

After algebraic simplification

$$\begin{gathered} \int x^4{(x^3+1)}^{2}dx=\int x^4\left\{{\left(x^3\right)}^2+2\times 1\times x^3+1\right\}dx \\ \int x^4{(x^3+1)}^{2}dx=\int x^4\left\{x^6+2x^3+1\right\}dx \\ \int x^4{(x^3+1)}^{2}dx=\int x^4{x}^6+2x^3{x}^4+1·x^{4}dx \\ \int x^4{(x^3+1)}^{2}dx=\int x^{10}+2x^7+x^{4}dx \end{gathered}$$

Now we can anti-differentiate immediately.

Step 4. The third integrals in Exercises 21–70 that we can anti differentiate immediately after algebraic simplification.

$\int \frac{2-e^x}{e^x}\mathrm{d}x$

After algebraic simplification.

$\int \frac{2-e^x}{e^x}\mathrm{d}x=\int \left(\frac{2}{e^x}-\frac{e^x}{e^x}\right)\mathrm{d}x$

Now we can anti-differentiate immediately.