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

$(2xe^{x^{2}})dx=?,\int (2xe^{x^{2}})dx=?$

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

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

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

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

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

5. Evaluate the indefinite integral

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

using the substitution method.

Solution
let $u=({\sqrt {x}})$ and $du=({\frac {1}{2({\sqrt {x}})}})dx$ to get, ${2}\int {cos(u)}du$ which is equal to ${2}{sin(u)}+C$ Substitute $u={\sqrt {x}}$ to get the answer $\int {\frac {\cos({\sqrt {x}})}{\sqrt {x}}}dx={{2}sin({\sqrt {x}})}+C$

6. Evaluate the indefinite integral

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

using the substitution method.

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

7. Evaluate the definite integral

$\int _{0}^{5}{\sqrt {3+2x}}dx$

using the substitution method.

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

8. Evaluate the definite integral

$\int _{0}^{\sqrt {\pi }}x\cos(x^{2})dx$

using the substitution method.

Solution
let $u=x^{2}$ and $du=(2x)dx$ to get, ${\frac {1}{2}}\int _{0}^{\sqrt {\pi }}cos(u)du$ which is equal to ${\frac {sin(u)}{2}}\|$ from $(0)to({\sqrt {\pi }})$ Substitute $u=x^{2}$ to get ${\frac {sin(x^{2})}{2}}$ then solve the integral to get the answer $\int _{0}^{\sqrt {\pi }}x\cos(x^{2})dx=0$

9. Evaluate the integral

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