Lecture 11

Faculty of Science; Department of Mathematics
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Readings For This Lecture

Group 5: Add a summary of the lecture in this space. Include examples, discussion, and links to external sources, if desired.

1. Calculate the differential of $y = (1 + x^{2})^{3} \cos (x^{2})^{3}$

2. Use differentials and the fundamental theorem of calculus to solve the indefinite integral

$(2 x e^{x^{2}}) d x = ?, \int (2 x e^{x^{2}}) d x = ?$

Solution
let $u = (x^{2})$ and $d u = (2 x) d x$ to get, $\int e^{u} d u$ which is equal to $e^{u} + C$ Substitute $u = x^{2}$ to get the answer $\int (2 x e^{x^{2}}) d x = e^{x^{2}} + C$

3. Use differentials and the fundamental theorem of calculus to solve the indefinite integral

$(3 x^{2} \cos (x^{3})) d x = ?, \int (3 x^{2} \cos (x^{3})) d x = ?$

4. Use differentials and the fundamental theorem of calculus to solve the indefinite integral

$(\frac{2 x}{1 + x^{4}}) d x = ?, \int (\frac{2 x}{1 + x^{4}}) d x = ?$

5. Evaluate the indefinite integral

$\int \frac{\cos (\sqrt{x})}{\sqrt{x}} d x$

using the substitution method.

Solution
let $u = (\sqrt{x})$ and $d u = (\frac{1}{2 (\sqrt{x})}) d x$ to get, $2 \int c o s (u) d u$ which is equal to $2 s i n (u) + C$ Substitute $u = \sqrt{x}$ to get the answer $\int \frac{\cos (\sqrt{x})}{\sqrt{x}} d x = 2 s i n (\sqrt{x}) + C$

6. Evaluate the indefinite integral

$\int x^{3} \sqrt{x^{4} + 1} d x$

using the substitution method.

Solution
let $u = (x^{4} + 1)$ then $\frac{d u}{4} = x^{3} d x$ . $\frac{1}{4} \int \sqrt{u} d u = \frac{1}{4} \frac{2}{3} u^{(\frac{3}{2})}$ sub original back into u $= \frac{1}{6} (x^{4} + 1)^{\frac{3}{2}}$

7. Evaluate the definite integral

$\int_{0}^{5} \sqrt{3 + 2 x} d x$

using the substitution method.

Solution
let $u = (3 + 2 x)$ then $(d u) = 2 d x$ . $\frac{1}{2} \int \sqrt{u} d u = \frac{1}{3} u^{(\frac{3}{2})}$ sub original back into u $= \frac{1}{3} (2 x + 3)^{\frac{3}{2}} \|_{x = 0}^{5}$ Evaluate integral to get: $\frac{13 \sqrt{(} 13)}{3} - \sqrt{3}$

8. Evaluate the definite integral

$\int_{0}^{\sqrt{π}} x \cos (x^{2}) d x$

using the substitution method.

Solution
let $u = x^{2}$ and $d u = (2 x) d x$ to get, $\frac{1}{2} \int_{0}^{\sqrt{π}} c o s (u) d u$ which is equal to $\frac{s i n (u)}{2} \|$ from $(0) t o (\sqrt{π})$ Substitute $u = x^{2}$ to get $\frac{s i n (x^{2})}{2}$ then solve the integral to get the answer $\int_{0}^{\sqrt{π}} x \cos (x^{2}) d x = 0$

9. Evaluate the integral

$\int x^{2} (x^{3} + 1)^{6} d x$ .