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Integral Calculus

Fundamental Theorem of Calculus

$$\int_a^b f(x)\,dx = F(b)-F(a)\quad \text{where }F'(x)=f(x)$$

Linearity of Indefinite Integrals

$$\int \bigl(f(x)\pm g(x)\bigr)\,dx=\int f(x)\,dx \pm \int g(x)\,dx$$
$$\int kf(x)\,dx = k\int f(x)\,dx$$
$$\int c\,dx = cx + C$$

Power and Logarithmic Integrals

$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad (n\ne -1)$$
$$\int \frac{dx}{x}=\ln|x|+C$$

Exponential Integrals

$$\int e^x\,dx=e^x+C$$
$$\int a^x\,dx=\frac{a^x}{\ln a}+C\quad (a>0,\;a\ne 1)$$

Definite Integral Properties

$$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$$
$$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$$
$$\int_a^b kf(x)\,dx = k\int_a^b f(x)\,dx$$
$$\int_a^b \bigl(f(x)\pm g(x)\bigr)\,dx=\int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx$$

Substitution Rule

$$u=g(x),\;du=g'(x)\,dx\quad\Rightarrow\quad \int f(g(x))\,g'(x)\,dx=\int f(u)\,du$$
$$\int_a^b f(g(x))\,g'(x)\,dx=\int_{u(a)}^{u(b)} f(u)\,du$$

Integration by Parts

$$\int u\,dv = uv - \int v\,du$$
$$\int_a^b u\,dv=\Big[uv\Big]_a^b-\int_a^b v\,du$$

Product-to-Sum Identities

$$2\sin A\sin B=\cos(A-B)-\cos(A+B)$$
$$2\cos A\cos B=\cos(A-B)+\cos(A+B)$$
$$2\sin A\cos B=\sin(A-B)+\sin(A+B)$$

Power-Reduction and Pythagorean Identities

$$\sin^2 x=\frac{1-\cos 2x}{2}$$
$$\cos^2 x=\frac{1+\cos 2x}{2}$$
$$1+\tan^2 x=\sec^2 x$$
$$1+\cot^2 x=\csc^2 x$$

Basic Trigonometric Integrals

$$\int \sin u\,du = -\cos u + C$$
$$\int \cos u\,du = \sin u + C$$
$$\int \sec^2 u\,du = \tan u + C$$
$$\int \csc^2 u\,du = -\cot u + C$$
$$\int \sec u\,\tan u\,du = \sec u + C$$
$$\int \csc u\,\cot u\,du = -\csc u + C$$
$$\int \tan u\,du = \ln|\sec u| + C$$
$$\int \cot u\,du = \ln|\sin u| + C$$
$$\int \sec u\,du = \ln|\sec u+\tan u| + C$$
$$\int \csc u\,du = \ln|\csc u-\cot u| + C$$

Inverse Trigonometric Forms

$$\int \frac{du}{a^2+u^2}=\frac{1}{a}\tan^{-1}\!\left(\frac{u}{a}\right)+C$$
$$\int \frac{du}{\sqrt{a^2-u^2}}=\sin^{-1}\!\left(\frac{u}{a}\right)+C$$
$$\int \frac{du}{a^2-u^2}=\frac{1}{2a}\ln\left|\frac{a+u}{a-u}\right|+C$$
$$\int \frac{du}{\sqrt{u^2-a^2}}=\ln\left|u+\sqrt{u^2-a^2}\right|+C$$

Trigonometric Substitution

Case 1: Expressions of the Form $\sqrt{a^2 - x^2}$

$$x = a\sin\theta$$
$$dx = a\cos\theta\,d\theta$$
$$\sqrt{a^2 - x^2} = a\cos\theta$$

Case 2: Expressions of the Form $\sqrt{a^2 + x^2}$

$$x = a\tan\theta$$
$$dx = a\sec^2\theta\,d\theta$$
$$\sqrt{a^2 + x^2} = a\sec\theta$$

Case 3: Expressions of the Form $\sqrt{x^2 - a^2}$

$$x = a\sec\theta$$
$$dx = a\sec\theta\tan\theta\,d\theta$$
$$\sqrt{x^2 - a^2} = a\tan\theta$$

Back-Substitution Using Right Triangles

$$\sin\theta=\frac{x}{a}\qquad \cos\theta=\frac{\sqrt{a^2-x^2}}{a}$$
$$\tan\theta=\frac{x}{a}\qquad \sec\theta=\frac{\sqrt{a^2+x^2}}{a}$$
$$\sec\theta=\frac{x}{a}\qquad \tan\theta=\frac{\sqrt{x^2-a^2}}{a}$$

Hyperbolic Functions

$$\sinh x=\frac{e^x-e^{-x}}{2}$$
$$\cosh x=\frac{e^x+e^{-x}}{2}$$
$$\tanh x=\frac{\sinh x}{\cosh x}$$
$$\coth x=\frac{\cosh x}{\sinh x}$$
$$\operatorname{sech}x=\frac{1}{\cosh x}$$
$$\operatorname{csch}x=\frac{1}{\sinh x}$$

Key Hyperbolic Identities

$$\cosh^2 x-\sinh^2 x=1$$
$$1-\tanh^2 x=\operatorname{sech}^2 x$$
$$\coth^2 x-\operatorname{csch}^2 x=1$$

Derivatives of Hyperbolic Functions

$$\frac{d}{dx}\bigl(\sinh u\bigr)=\cosh u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\cosh u\bigr)=\sinh u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\tanh u\bigr)=\operatorname{sech}^2 u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\coth u\bigr)=-\operatorname{csch}^2 u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\operatorname{sech}u\bigr)=-\operatorname{sech}u\,\tanh u\,\frac{du}{dx}$$
$$\frac{d}{dx}\bigl(\operatorname{csch}u\bigr)=-\operatorname{csch}u\,\coth u\,\frac{du}{dx}$$

Improper Integrals

$$\int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dx$$
$$\int_{-\infty}^b f(x)\,dx=\lim_{a\to-\infty}\int_a^b f(x)\,dx$$
$$\int_a^b f(x)\,dx=\lim_{t\to c^-}\int_a^t f(x)\,dx+\lim_{t\to c^+}\int_t^b f(x)\,dx$$

Applications: Area Under a Curve

$$A=\int_a^b f(x)\,dx$$

Applications: Area Between Curves

$$A=\int_a^b \bigl(f(x)-g(x)\bigr)\,dx$$
$$A=\int_c^d \bigl(F(y)-G(y)\bigr)\,dy$$

Applications: Volume by Cross-Sections

$$V=\int_a^b A(x)\,dx$$

Applications: Volume of Revolution (Disk)

$$V=\pi\int_a^b \bigl(R(x)\bigr)^2\,dx$$

Applications: Volume of Revolution (Washer)

$$V=\pi\int_a^b \Big(\bigl(R(x)\bigr)^2-\bigl(r(x)\bigr)^2\Big)\,dx$$

Applications: Volume of Revolution (Shell)

$$V=2\pi\int_a^b \bigl(\text{radius}\bigr)\bigl(\text{height}\bigr)\,dx$$
$$V=2\pi\int_c^d \bigl(\text{radius}\bigr)\bigl(\text{height}\bigr)\,dy$$

Applications: Arc Length

$$L=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$$
$$L=\int_c^d \sqrt{1+\left(\frac{dx}{dy}\right)^2}\,dy$$

Applications: Surface Area of Revolution

$$S=2\pi\int_a^b y\,\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$$
$$S=2\pi\int_a^b x\,\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$$

Pappus' Centroid Theorem I (Surface Area of Revolution)

$$S=2\pi d\,L$$
$$S=2\pi \bar{r}\,L$$

Pappus' Centroid Theorem II (Volume of Revolution)

$$V=2\pi d\,A$$
$$V=2\pi \bar{r}\,A$$

Centroid Distance to Axis

$$d=\bar{r}$$

Area Moments of Inertia (Second Moments of Area)

$$I_x=\int y^2\,dA$$
$$I_y=\int x^2\,dA$$
$$I_{xy}=\int xy\,dA$$

Polar Moment of Inertia

$$J_O=\int r^2\,dA$$
$$J_O=I_x+I_y$$

Parallel-Axis Theorem (Area)

$$I_x=I_{\bar{x}}+A d^2$$
$$I_y=I_{\bar{y}}+A d^2$$

Parallel-Axis Theorem (Polar)

$$J_O=J_C+A d^2$$

Radius of Gyration

$$k_x=\sqrt{\frac{I_x}{A}}$$
$$k_y=\sqrt{\frac{I_y}{A}}$$
$$k_O=\sqrt{\frac{J_O}{A}}$$

Mass Moment of Inertia

$$I=\int r^2\,dm$$
$$I_x=\int (y^2+z^2)\,dm$$
$$I_y=\int (x^2+z^2)\,dm$$
$$I_z=\int (x^2+y^2)\,dm$$

Parallel-Axis Theorem (Mass)

$$I_O=I_C+M d^2$$

Applications: Work

$$W=\int_a^b F(x)\,dx$$
$$W=\int_C \vec{F}\cdot d\vec{r}$$
$$W=\int_a^b F(x)\,\frac{dx}{dt}\,dt$$

Applications: Pumping Work

$$dW=\rho g\,A(y)\,d(y)\,dy$$
$$W=\rho g\int_{y_1}^{y_2} A(y)\,d(y)\,dy$$

Applications: Hydrostatic Pressure

$$p=\rho g h$$

Applications: Hydrostatic Force on a Vertical Plate

$$F=\int p\,dA$$
$$F=\rho g\int h\,dA$$
$$F=\rho g\int_{y_1}^{y_2} h(y)\,w(y)\,dy$$

Applications: Center of Mass (Lamina)

$$M=\iint_R \rho(x,y)\,dA$$
$$\bar{x}=\frac{1}{M}\iint_R x\,\rho(x,y)\,dA$$
$$\bar{y}=\frac{1}{M}\iint_R y\,\rho(x,y)\,dA$$
$$\text{If }\rho\text{ is constant: }M=\rho A,\;\bar{x}=\frac{1}{A}\iint_R x\,dA,\;\bar{y}=\frac{1}{A}\iint_R y\,dA$$

Applications: Centroid Under $y=f(x)$

$$A=\int_a^b f(x)\,dx$$
$$\bar{x}=\frac{1}{A}\int_a^b x\,f(x)\,dx$$
$$\bar{y}=\frac{1}{A}\cdot\frac{1}{2}\int_a^b \bigl(f(x)\bigr)^2\,dx$$

Applications: Centroid Between $y=f(x)$ and $y=g(x)$

$$A=\int_a^b \bigl(f(x)-g(x)\bigr)\,dx$$
$$\bar{x}=\frac{1}{A}\int_a^b x\bigl(f(x)-g(x)\bigr)\,dx$$
$$\bar{y}=\frac{1}{A}\cdot\frac{1}{2}\int_a^b \Big(\bigl(f(x)\bigr)^2-\bigl(g(x)\bigr)^2\Big)\,dx$$

Average Value of a Function

$$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx$$
Concept

Basic Integrals

1. $$\int (8x^3 - 2x + \sqrt{5})\,dx$$

2. $$\int (3 - 2x)(4x + 5)\,dx$$

3. $$\int \frac{(x - 2)^3}{3x^2}\,dx$$

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

5. $$\int \sqrt{x^{1/2} - 2x^{3/2} + x^{5/2}}\,dx$$

6. $$\int \left(\frac{4}{x^2} - \frac{3}{x^3} + \frac{2}{x^4}\right)\,dx$$

7. $$\int \sqrt[3]{t}(\sqrt{t} + 1)^2\,dt$$

8. $$\int \frac{(5 - 2\sqrt{t})^2}{\sqrt[4]{t^3}}\,dt$$

9. $$\int \frac{x^{4/3} - 16}{x^{1/3} + 2}\,dx$$

10. $$\int \left(4 - \frac{3}{\sqrt{x}}\right)^2 \frac{dx}{\sqrt{x}}$$

11. $$\int (8x^3 - 3x^2 + 5x + 3)\,dx$$

12. $$\int_{2}^{3} \left(1 - \frac{x}{2}\right)^2\,dx$$

13. $$\int_{2}^{8} \frac{dy}{y\sqrt{2y}}$$

14. $$\int_{1}^{9} \sqrt{x(4 - x)}\,dx$$

15. $$\int_{1}^{4} \left(4x - \frac{5}{x\sqrt{x}}\right)\,dx$$

Integral Calculus – Basic Integrals – Diagram Integral Calculus – Basic Integrals – Diagram Integral Calculus – Basic Integrals – Diagram

1. $$\int (8x^3 - 2x + \sqrt{5})\,dx = 2x^4 - x^2 + \sqrt{5}\,x + C$$

2. $$\int (3 - 2x)(4x + 5)\,dx = -\frac{8}{3}x^3 + x^2 + 15x + C$$

3. $$\int \frac{(x - 2)^3}{3x^2}\,dx = \frac{x^2}{6} - 2x + 4\ln|x| + \frac{8}{3x} + C$$

4. $$\int (x^{3/4} + 2x^{1/2})^2\,dx = \frac{2}{5}x^{5/2} + \frac{16}{9}x^{9/4} + 2x^2 + C$$

5. $$\int \sqrt{x^{1/2} - 2x^{3/2} + x^{5/2}}\,dx = \int x^{1/4}|x-1|\,dx$$

$$= \begin{cases} \frac{4}{9}x^{9/4} - \frac{4}{5}x^{5/4} + C, & x\ge 1 \\ \frac{4}{5}x^{5/4} - \frac{4}{9}x^{9/4} + C, & 0\le x\le 1 \end{cases} $$

6. $$\int \left(\frac{4}{x^2} - \frac{3}{x^3} + \frac{2}{x^4}\right)\,dx = -\frac{4}{x} + \frac{3}{2x^2} - \frac{2}{3x^3} + C$$

7. $$\int \sqrt[3]{t}(\sqrt{t} + 1)^2\,dt = \frac{3}{7}t^{7/3} + \frac{12}{11}t^{11/6} + \frac{3}{4}t^{4/3} + C$$

8. $$\int \frac{(5 - 2\sqrt{t})^2}{\sqrt[4]{t^3}}\,dt = 100t^{1/4} - \frac{80}{3}t^{3/4} + \frac{16}{5}t^{5/4} + C$$

9. $$\int \frac{x^{4/3} - 16}{x^{1/3} + 2}\,dx = \frac{1}{2}x^2 - \frac{6}{5}x^{5/3} + 3x^{4/3} - 8x + C$$

10. $$\int \left(4 - \frac{3}{\sqrt{x}}\right)^2 \frac{dx}{\sqrt{x}} = 32\sqrt{x} - 24\ln x - \frac{18}{\sqrt{x}} + C$$

11. $$\int (8x^3 - 3x^2 + 5x + 3)\,dx = 2x^4 - x^3 + \frac{5}{2}x^2 + 3x + C$$

12. $$\int_{2}^{3} \left(1 - \frac{x}{2}\right)^2\,dx = \frac{1}{12}$$

13. $$\int_{2}^{8} \frac{dy}{y\sqrt{2y}} = \frac{1}{2}$$

14. $$\int_{1}^{9} \sqrt{x}(4 - x)\,dx = -\frac{412}{15}$$

15. $$\int_{1}^{4} \left(4x - \frac{5}{x\sqrt{x}}\right)\,dx = 25$$

Integral Calculus – Basic Integrals – Diagram Integral Calculus – Basic Integrals – Diagram Integral Calculus – Basic Integrals – Diagram Integral Calculus – Basic Integrals – Diagram

U-Substitution Problem Set

1. $$\int 5x(4 - 2x^2)^4\,dx$$

2. $$\int \frac{5x^2\,dx}{(1 - 2x^3)^6}$$

3. $$\int \frac{5y - 10}{y^2 - 4y + 11}\,dy$$

4. $$\int \frac{x\,dx}{4x^4 - 4x^2 + 1}$$

5. $$\int_{0}^{1} \frac{\tan^{-1} z}{z^2 + 1}\,dz$$

6. $$\int \frac{dz}{(2z - 3)^{3/2}}$$

7. $$\int (1 + 4e^{3x})^3 e^{3x}\,dx$$

8. $$\int \frac{2x^2 - 7x - 5}{2x + 1}\,dx$$

9. $$\int \frac{w + 1}{5w + 11}\,dw$$

10. $$\int \frac{x}{3x - 4}\,dx$$

11. $$\int \frac{x^5 - x^3 + 2x}{x^2 + 4}\,dx$$

12. $$\int \frac{e^{2x} - 1}{e^{2x} + 1}\,dx$$

13. $$\int \frac{dx}{x\ln^2 x}$$

14. $$\int_{0}^{\pi/2} \frac{\cos\beta\,d\beta}{\sqrt{1 + \sin\beta}}$$

15. $$\int \frac{\sin 2x\,dx}{\sqrt{\cos 2x - 1}}$$

16. $$\int \frac{dx}{x^{2/3}(\sqrt[3]{x}+5)^{3/2}}$$

17. $$\int \frac{e^x \sec^2(e^x + 1)}{\tan(e^x + 1)}\,dx$$

18. $$\int \frac{2e^{5x}}{3 - e^{5x}}\,dx$$

Integral Calculus – U-Substitution Problem Set – Diagram Integral Calculus – U-Substitution Problem Set – Diagram Integral Calculus – U-Substitution Problem Set – Diagram

1. $$\int 5x(4 - 2x^2)^4\,dx = -\frac{1}{4}(4 - 2x^2)^5 + C$$

2. $$\int \frac{5x^2}{(1 - 2x^3)^6}\,dx = \frac{1}{6(1 - 2x^3)^5} + C$$

3. $$\int \frac{5y - 10}{y^2 - 4y + 11}\,dy = \frac{5}{2}\ln(y^2 - 4y + 11) + C$$

4. $$\int \frac{x}{4x^4 - 4x^2 + 1}\,dx = \frac{1}{4}\ln(4x^4 - 4x^2 + 1) + C$$

5. $$\int_{0}^{1} \frac{\tan^{-1} z}{z^2 + 1}\,dz = \frac{\pi^2}{32}$$

6. $$\int \frac{dz}{(2z - 3)^{3/2}} = -\frac{1}{\sqrt{2z - 3}} + C$$

7. $$\int (1 + 4e^{3x})^3 e^{3x}\,dx = \frac{(1 + 4e^{3x})^4}{48} + C$$

8. $$\int \frac{2x^2 - 7x - 5}{2x + 1}\,dx = x^2 - 4x - \frac{1}{2}\ln|2x + 1| + C$$

9. $$\int \frac{w + 1}{5w + 11}\,dw = \frac{5w-6\ln(5w+11)}{25} + C$$

10. $$\int \frac{x}{3x - 4}\,dx = \frac{1}{3}x + \frac{4}{9}\ln|3x - 4| + C$$

11. $$\int \frac{x^5 - x^3 + 2x}{x^2 + 4}\,dx = \frac{1}{4}x^4 - \frac{5}{2}x^2 + 11\ln(x^2 + 4) + C$$

12. $$\int \frac{e^{2x} - 1}{e^{2x} + 1}\,dx = \ln(e^{2x} + 1) - x + C$$

13. $$\int \frac{dx}{x\ln^2 x} = -\frac{1}{\ln x} + C$$

14. $$\int_{0}^{\pi/2} \frac{\cos\beta}{\sqrt{1 + \sin\beta}}\,d\beta = 2(\sqrt{2} - 1)$$

15. $$\int \frac{\sin 2x}{\sqrt{\cos 2x - 1}}\,dx = -\sqrt{\cos 2x - 1} + C$$

16. $$\int \frac{dx}{x^{2/3}(\sqrt[3]{x}+5)^{3/2}} = -\frac{6}{\sqrt{\sqrt[3]{x}+5}} + C$$

17. $$\int \frac{e^x \sec^2(e^x + 1)}{\tan(e^x + 1)}\,dx = \ln|\tan(e^x + 1)| + C$$

18. $$\int \frac{2e^{5x}}{3 - e^{5x}}\,dx = -\frac{2}{5}\ln|e^{5x}-3| + C$$

Integral Calculus – U-Substitution Problem Set – Diagram Integral Calculus – U-Substitution Problem Set – Diagram Integral Calculus – U-Substitution Problem Set – Diagram Integral Calculus – U-Substitution Problem Set – Diagram

U-Substitution with Exponential Functions

1. $$\int 3e^{5x}\,dx$$

2. $$\int 3^{\tan(4x)}\sec^{2}(4x)\,dx$$

3. $$\int \frac{dy}{\sqrt[3]{e^{2y}}}$$

4. $$\int e^{4-3x}\,dx$$

5. $$\int \frac{e^{3x}}{e^{x}-1}\,dx$$

6. $$\int x^{-2}e^{1/x}\,dx$$

7. $$\int \frac{5^{x}}{e^{2x}}\,dx$$

8. $$\int e^{3x}\sqrt[3]{\,4-e^{3x}\,}\,dx$$

9. $$\int \frac{3-4e^{5x}}{e^{2x}}\,dx$$

10. $$\int \left(e^{2x}-e^{-3x}\right)^{2}\,dx$$

11. $$\int \frac{5^{\cot(x/3)}}{\sin^{2}(x/3)}\,dx$$

12. $$\int 8^{\,1-2x^{2}}\,x\,dx$$

13. $$\int_{1}^{3}\sqrt[3]{5^{2y}}\,dy$$

14. $$\int_{0}^{3}\left(4-e^{-z}\right)^{2}\,dz$$

15. $$\int_{0}^{2}\left(e^{2x}+xe\right)\,dx$$

16. $$\int \frac{5^{2x}}{(5^{2x}-1)^{3}}\,dx$$

Integral Calculus – U-Substitution with Exponential Functions – Diagram Integral Calculus – U-Substitution with Exponential Functions – Diagram Integral Calculus – U-Substitution with Exponential Functions – Diagram

1. $$\int 3e^{5x}\,dx=\frac{3}{5}e^{5x}+C$$

2. $$\int 3^{\tan(4x)}\sec^{2}(4x)\,dx=\frac{3^{\tan(4x)}}{4\ln 3}+C$$

3. $$\int \frac{dy}{\sqrt[3]{e^{2y}}}=-\frac{3}{2}e^{-2y/3}+C$$

4. $$\int e^{4-3x}\,dx=-\frac{1}{3}e^{4-3x}+C$$

5. $$\int \frac{e^{3x}}{e^{x}-1}\,dx=\frac{1}{2}e^{2x}+e^{x}+\ln|e^{x}-1|+C$$

6. $$\int x^{-2}e^{1/x}\,dx=-e^{1/x}+C$$

7. $$\int \frac{5^{x}}{e^{2x}}\,dx=\frac{5^{x}}{(\ln 5-2)e^{2x}}+C$$

8. $$\int e^{3x}\sqrt[3]{\,4-e^{3x}\,}\,dx=-\frac{1}{4}\left(4-e^{3x}\right)^{4/3}+C$$

9. $$\int \frac{3-4e^{5x}}{e^{2x}}\,dx=-\frac{3}{2}e^{-2x}-\frac{4}{3}e^{3x}+C$$

10. $$\int \left(e^{2x}-e^{-3x}\right)^{2}\,dx=\frac{1}{4}e^{4x}+2e^{-x}-\frac{1}{6}e^{-6x}+C$$

11. $$\int \frac{5^{\cot(x/3)}}{\sin^{2}(x/3)}\,dx=-\frac{3}{\ln 5}\,5^{\cot(x/3)}+C$$

12. $$\int 8^{\,1-2x^{2}}\,x\,dx=-\frac{8^{\,1-2x^{2}}}{4\ln 8}+C$$

13. $$\int_{1}^{3}\sqrt[3]{5^{2y}}\,dy=\frac{3}{2\ln 5}\left(25-5^{2/3}\right)$$

14. $$\int_{0}^{3}\left(4-e^{-z}\right)^{2}\,dz=\frac{81}{2}+8e^{-3}-\frac{1}{2}e^{-6}$$

15. $$\int_{0}^{2}\left(e^{2x}+xe\right)\,dx=\frac{e^{4}-1}{2}+2e$$

16. $$\int \frac{5^{2x}}{(5^{2x}-1)^{3}}\,dx=-\frac{1}{4\ln 5}\cdot\frac{1}{(5^{2x}-1)^{2}}+C$$

Integral Calculus – U-Substitution with Exponential Functions – Diagram Integral Calculus – U-Substitution with Exponential Functions – Diagram Integral Calculus – U-Substitution with Exponential Functions – Diagram Integral Calculus – U-Substitution with Exponential Functions – Diagram

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Integral Calculus – Problem: – Diagram Integral Calculus – Problem: – Diagram Integral Calculus – Problem: – Diagram Integral Calculus – Problem: – Diagram

Problem:

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Integral Calculus – Problem: – Diagram Integral Calculus – Problem: – Diagram Integral Calculus – Problem: – Diagram Integral Calculus – Problem: – Diagram
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