diff --git a/_freeze/presentations/04b-derivatives/figure-beamer/cell-2-output-1.pdf b/_freeze/presentations/04b-derivatives/figure-beamer/cell-2-output-1.pdf index 29b2228..2cfe183 100644 Binary files a/_freeze/presentations/04b-derivatives/figure-beamer/cell-2-output-1.pdf and b/_freeze/presentations/04b-derivatives/figure-beamer/cell-2-output-1.pdf differ diff --git a/presentations/08b-integration-techniques.qmd b/presentations/08b-integration-techniques.qmd index 08b0458..c010d1e 100644 --- a/presentations/08b-integration-techniques.qmd +++ b/presentations/08b-integration-techniques.qmd @@ -48,24 +48,27 @@ Example: $\displaystyle \int_0^2 (1+x)^5 \, dx$ ## Examples -Compute the following integrals: +$\displaystyle \int a^{bx} \, dx$ -- $\displaystyle \int a^{bx} \, dx$ -- $\displaystyle \int x^3 \cos(x^4 + 2) \, dx$ +## Examples + +$\displaystyle \int x^3 \cos(x^4 + 2) \, dx$ ## Examples -Compute the following integrals: +$\displaystyle \int \frac{\sin(3 \ln x)}{x} \, dx$ + +## Examples -- $\displaystyle \int \frac{\sin(3 \ln x)}{x} \, dx$ -- $\displaystyle \int x^5 \sqrt{1 + x^2} \, dx$ +$\displaystyle \int x^5 \sqrt{1 + x^2} \, dx$ ## Examples -Compute the following integrals: +$\displaystyle \int e^x \sqrt{1 + e^x} \, dx$ -- $\displaystyle \int e^x \sqrt{1 + e^x} \, dx$ -- $\displaystyle \int \frac{dx}{x^2 + 4x + 5}$ +## Examples + +$\displaystyle \int \frac{dx}{x^2 + 4x + 5}$ ## Trigonometric integration @@ -78,27 +81,39 @@ Compute the following integrals: ## Examples -Compute the following integrals: +$\displaystyle \int \sec x \, dx$ -- $\displaystyle \int \sec x \, dx$ -- $\displaystyle \int \csc x \, dx$ +## Examples + +$\displaystyle \int \csc x \, dx$ ## Examples -Compute the following integrals: +$\displaystyle \int \sin^3 x \cos^8 x \, dx$ + +## Examples -- $\displaystyle \int \sin^3 x \cos^8 x \, dx$ -- $\displaystyle \int \cos^5 x \, dx$ -- $\displaystyle \int \cos^2 x \, dx$ -- $\displaystyle \int \sin^2 x \, dx$ -- $\displaystyle \int \sin^4 x \, dx$ +$\displaystyle \int \cos^5 x \, dx$ ## Examples -Compute the following integrals: +$\displaystyle \int \cos^2 x \, dx$ -- $\displaystyle \int \tan^2 x \, dx$ -- $\displaystyle \int \sec^4 x \, dx$ +## Examples + +$\displaystyle \int \sin^2 x \, dx$ + +## Examples + +$\displaystyle \int \sin^4 x \, dx$ + +## Examples + +$\displaystyle \int \tan^2 x \, dx$ + +## Examples + +$\displaystyle \int \sec^4 x \, dx$ ## Integration by parts @@ -110,10 +125,11 @@ Compute the following integrals: ## Examples -Compute the following integrals: +$\displaystyle \int \sec^3 x \, dx$ + +## Examples -- $\displaystyle \int \sec^3 x \, dx$ -- $\displaystyle \int e^{ax} \cos(bx) \, dx$ +$\displaystyle \int e^{ax} \cos(bx) \, dx$ ## Reduction formula @@ -125,7 +141,9 @@ Compute the following integrals: Let $\displaystyle I_n = \int_0^{\pi/2} \cos^n x \, dx$. Find a reduction formula for $I_n$. -## Solutions +# Solutions + +## - $\displaystyle \int x \sin(1 + x^2) \, dx = - \frac{1}{2} \cos(1 + x^2) + C$ - $\displaystyle \int_0^2 (1 + x)^5 \, dx = \frac{364}{3}$