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« Integration Formulas
Introduction to Integration: (lesson 2 of 2)

Integration Formulas - Exercises


1. Integration of polynomial functions


Formula 1.

∫xrdx=xr+1r+1+C \color{blue}{ \int x^r dx = \frac{x^{r+1}}{r+1}+C}

Exercise 1.

∫x7dx=x7+17+1+C=x88+C \int x^{\color{red}{7}} dx = \frac{ x^{\color{red}{7}+1} }{\color{red}{7}+1} + C = \frac{x^8}{8}+C

Try yourself
∫x12 dx= \color{blue}{\int x^{12} \,dx = }
x1111+C \frac{x^{11}}{11} + C x1212+C \frac{x^{12}}{12} + C
x1313+C \frac{x^{13}}{13} + C x1414+C \frac{x^{14}}{14} + C

Formula 2.

∫k dx=k x+C  where  k  is constant. \color{blue}{ \int k \, dx = k\,x + C } ~ \text{ where } ~ k ~ \text{ is constant.}

Exercise 2.

∫5dx=5 x+C \int \color{blue}{5} dx = \color{blue}{5}\,x + C

Formula 3.

∫k⋅f(x)dx=k∫f(x)dx+C  where  k  is constant. \color{blue}{ \int k \cdot f(x) dx = k \int f(x) dx + C } ~ \text{ where } ~ k ~ \text{ is constant.}

Exercise 3.

∫4x6dx=4∫x6dx=4⋅x6+16+1+C=47x7+C \int \color{blue}{4} x^6 dx = \color{blue}{4} \int x^6 dx = 4 \cdot \frac{x^{6+1}}{6+1} + C = \frac{4}{7} x^7 + C

Formula 4.

∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx \color{blue}{ \int f(x) \pm g(x) dx = \int f(x) dx \pm \int g(x) dx}

Exercise 4.

∫2x2+3x+2dx=∫2x2dx+∫3xdx+∫2dx==2x2+12+1+3x1+11+1+2x+C==23x3+32x2+2x+C \begin{aligned} \int \color{red}{2x^2} + \color{blue}{3x} + 2 dx &= \int \color{red}{2x^2} dx + \int \color{blue}{3x} dx + \int 2 dx = \\\\ &= 2 \color{red}{\frac{x^{2+1}}{2+1}} + 3 \color{blue}{\frac{x^{1+1}}{1+1}} + 2x + C = \\\\ &= \frac{2}{3} x^3 + \frac{3}{2} x^2 + 2x + C \end{aligned}

Try yourself
∫3x2+3x−1 dx= \color{blue}{\int 3x^2 + 3x-1 \,dx = }
x3+32x2−x+C x^3 + \frac{3}{2}x^2 - x + C x3−32x2−x+C x^3 - \frac{3}{2}x^2 - x + C
x3−32x2+x+C x^3 - \frac{3}{2}x^2 + x + C x3+3x2+x2+C x^3 + 3x^2 + x^2 + C

2. Integration of exponential and logarithmic functions

Formula 5.

∫exdx=ex+C \color{blue}{ \int e^x dx = e^x + C}

Exercise 5.

∫6exdx=6∫exdx=6ex+C \int 6 e^x dx = 6 \int e^x dx = 6 e^x + C

Formula 6.

∫ef(x)⋅f′(x)dx=ef(x)+C \color{blue}{ \int e^{f(x)} \cdot f'(x) dx = e^{f(x)} + C}

Exercise 6.

∫e2x2+1⋅4xdx=∫e2x2+1⋅(2x2+1)′dx==e2x2+1+C \begin{aligned} \int e^{2x^2+1}\cdot \color{blue}{4x} dx &= \int e^{2x^2+1}\cdot \color{blue}{\left(2x^2+1\right)'} dx = \\\\ &= e^{2x^2+1}+C \end{aligned}

Try yourself
∫4x3ex4 dx= \color{blue}{\int 4 x^3 e^{x^4} \,dx = }
x⋅ex3+C x \cdot e^{x^3} + C x⋅ex4+C x \cdot e^{x^4} + C
ex3+C e^{x^3} + C ex4+C e^{x^4} + C

Formula 7.

∫1xdx=ln⁡∣x∣+C \color{blue}{ \int \frac{1}{x} dx = \ln|x| + C}

Exercise 7.

∫5xdx=5∫1xdx=5 ln⁡∣x∣+C \int \frac{5}{x} dx = 5 \int \frac{1}{x} dx = 5\,\ln|x| + C

Formula 8.

∫f′(x)f(x)=ln⁡ ∣f(x)∣+C \color{blue}{ \int \frac{f'(x)}{f(x)} = \ln\,|f(x)| + C }

Exercise 8a.

∫2xx2−1dx=∫(x2−1)′x2−1=ln⁡∣x2−1∣+C \int \frac{\color{blue}{2x}}{x^2-1} dx = \int \frac{\color{blue}{(x^2-1)'}}{x^2-1} = \ln|x^2-1| + C

Exercise 8b.

∫x2+1x3+3x+2dx=∫13(3x2+3)x3+3x+2dx==13∫(x3+3x+2)′x3+3x+2dx==13ln⁡∣x3+3x+2∣+C \begin{aligned} \int \frac{x^2+1}{x^3+3x+2} dx &= \int \frac{\color{blue}{\frac{1}{3}(3x^2 + 3)}}{x^3+3x+2} dx = \\\\ &= \frac{1}{3} \int \frac{\color{blue}{(x^3+3x+2)'}}{x^3+3x+2} dx = \\\\ &= \frac{1}{3} \ln|x^3+3x+2| + C \end{aligned}

Try yourself
∫x1+x2 dx= \color{blue}{\int \frac{x}{1+x^2} \,dx = }
ln⁡∣1+x2∣+C \ln \left| 1 + x^2 \right| + C ln⁡∣1+x∣+C \ln \left| 1 + x \right| + C
12ln⁡∣1+x2∣+C \frac{1}{2} \ln \left| 1 + x^2 \right| + C 2ln⁡∣1+x2∣+C 2 \ln \left| 1 + x^2 \right| + C


3. Integration of trigonometric functions

Formula 9.

∫cos⁡x dx=sin⁡x+C \color{blue}{ \int \cos x \, dx = \sin x + C}

Exercise 9.

∫3cos⁡x dx=3∫cos⁡x dx=3sin⁡x+C \int 3 \cos x \, dx = 3 \int \cos x \, dx = 3 \sin x + C

Formula 10.

∫cos⁡f(x)⋅f′(x)dx=sin⁡f(x)+C \color{blue}{ \int \cos f(x) \cdot f'(x) dx = \sin f(x) + C}

Exercise 10.

∫x2cos⁡(x3)dx=13∫3x2cos⁡(x3)dx==13∫(x3)′cos⁡(x3)dx==13sin⁡(x3)+C \begin{aligned} \int x^2 \cos(x^3) dx &= \frac{1}{3} \int \color{blue}{3x^2} \cos(x^3) dx = \\\\ &= \frac{1}{3} \int \color{blue}{\left( x^3 \right)'} \cos(x^3) dx = \\\\ &= \frac{1}{3} \sin(x^3) + C \end{aligned}

Try yourself
∫4xcos⁡(x2) dx= \color{blue}{\int 4x \cos\left(x^2\right) \,dx = }
2xsin⁡(x2)+C 2x\sin\left(x^2\right)+C 2sin⁡(x2)+C 2\sin\left(x^2\right)+C
xsin⁡(x2)+C x\sin\left(x^2\right)+C sin⁡(x2)+C \sin\left(x^2\right)+C

Formula 11.

∫sin⁡x dx=−cos⁡x+C \color{blue}{ \int \sin x \, dx = - \cos x + C}

Exercise 11.

∫−2sin⁡x dx=−2∫sin⁡x dx=−2(−cos⁡x)+C=2cos⁡x+C \int -2 \sin x \, dx = -2 \int \sin x \, dx = -2(- \cos x) + C = 2\cos x + C

Formula 12.

∫sin⁡f(x)⋅f′(x)dx=−cos⁡f(x)+C \color{blue}{ \int \sin f(x) \cdot f'(x) dx = - \cos f(x) + C }

Exercise 12.

∫xsin⁡(1+x2)dx=12∫2xsin⁡(1+x2)dx==12∫(1+x2)′sin⁡(1+x2)dx==−12cos⁡(1+x2)+C \begin{aligned} \int x \sin \left(1+x^2\right) dx &= \frac{1}{2} \int \color{blue}{2x} \sin \left(1+x^2\right) dx = \\\\ &= \frac{1}{2} \int \color{blue}{\left(1+x^2\right)'} \sin \left(1+x^2\right) dx = \\\\ &= - \frac{1}{2} \cos \left( 1+x^2 \right) + C \end{aligned}

Try yourself
∫6x2sin⁡(x3+2) dx= \color{blue}{\int 6x^2 \sin\left(x^3+2\right) \,dx = }
sin⁡(x3+2)+C \sin\left(x^3+2\right)+C cos⁡(x3+2)+C \cos\left(x^3+2\right)+C
−2sin⁡(x3+2)+C -2\sin\left(x^3+2\right)+C −2cos⁡(x3+2)+C -2\cos\left(x^3+2\right)+C

Formula 13.

∫sec⁡2xdx=tan⁡x+C \color{blue}{ \int \sec^2 x dx = \tan x + C}

Exercise 13.

∫9sec⁡2xdx=9∫sec⁡2xdx=9tan⁡x+C \int 9 \sec^2 x dx = 9 \int \sec^2 x dx = 9 \tan x + C

Formula 14.

∫sec⁡2f(x)⋅f′(x)dx=tan⁡f(x)+C \color{blue}{ \int \sec^2 f(x) \cdot f'(x) dx = \tan f(x) + C }

Exercise 14.

∫x3⋅sec⁡2(1+x4)=14∫4x3sec⁡2(1+x4)dx==14∫(1+x4)′sec⁡2(1+x4)dx==14tan⁡(1+x4)+C \begin{aligned} \int x^3 \cdot \sec^2 \left(1+x^4\right) &= \frac{1}{4} \int \color{blue}{4x^3} \sec^2 \left(1+x^4\right) dx = \\\\ &= \frac{1}{4} \int \color{blue}{\left(1+x^4\right)'} \sec^2 \left(1+x^4\right) dx = \\\\ &= \frac{1}{4} \tan \left(1+x^4\right) + C \end{aligned}


4. Integration by substitution

Formula 15.

∫f(u)du=F(u)+C where u=g(x) and du=g′(x)dx \color{blue}{ \int f(u) du = F(u) + C} ~ \text{where} ~ u=g(x) ~ \text{and} ~ du=g'(x)dx

Also written as:

∫f(g(x))⋅g′(x)dx=F(g(x))+C \int f(g(x)) \cdot g'(x) dx = F(g(x)) + C

Exercise 15a:

∫2x⋅(x2+4)4dx \int 2x \cdot \left( x^2+4\right)^4 dx
u=x2+4du=2xdx \begin{aligned} \color{red}{u} & \color{red}{=x^2 + 4} \\\\ \color{blue}{du}& \color{blue}{= 2x dx} \end{aligned}
∫2x⋅(x2+4)4dx=∫u4du==u55+C==(x2+4)25+C \begin{aligned} \int \color{blue}{2x} \cdot \color{red}{\left( x^2+4\right)^4} \color{blue}{dx} &= \int \color{red}{u^4} \color{blue}{du}= \\\\ &= \frac{u^5}{5} + C = \\\\ &= \frac{\left(x^2+4\right)^2}{5} + C \end{aligned}

Exercise 15b:

∫x2⋅x3+1dx \int x^2 \cdot \sqrt{x^3+1} dx
u=x3+1du=3x2dx \begin{aligned} \color{red}{u} & \color{red}{=x^3 + 1} \\\\ \color{blue}{du}& \color{blue}{= 3x^2 dx} \end{aligned}
∫x2x3+1dx=13∫x3+1 3x2dx==13∫u  du==13∫u1\2 du=13u3\232+C=29u3\2+C=29(x3+1)3\2+C=29(x3+1)3+C \begin{aligned} \int x^2 \sqrt{x^3+1} dx &= \frac{1}{3} \int \sqrt{\color{red}{x^3+1}} \, \color{blue}{3x^2 dx} = \\\\ &= \frac{1}{3} \int \sqrt{\color{red}{u}} \, \color{blue}{\, du} = \\\\ &= \frac{1}{3} \int u^{1 \backslash 2} \, du \\\\ &= \frac{1}{3} \frac{ u^{3 \backslash 2 }}{\frac{3}{2}} + C \\\\ &= \frac{2}{9} u^{3 \backslash 2}+C \\\\ &= \frac{2}{9} (x^3+1)^{3 \backslash 2}+C \\\\ &= \frac{2}{9} \sqrt{\left(x^3+1\right)^3} + C \end{aligned}

Try yourself
∫xx2−1 dx= \color{blue}{\int x \sqrt{x^2-1} \,dx = }
13(x2−1)3\2 \frac{1}{3} \left(x^2-1\right)^{3 \backslash 2} 3(x2−1)3\2 3 \left(x^2-1\right)^{3 \backslash 2}
12(x2−1)3\2 \frac{1}{2} \left(x^2-1\right)^{3 \backslash 2} 2(x2−1)3\2 2 \left(x^2-1\right)^{3 \backslash 2}


5. Integration by parts

Formula 16.

∫f(x)g′(x)dx=f(x)g(x)−∫f′(x)g(x)dx \color{blue}{ \int f(x)g'(x) dx = f(x)g(x) - \int f'(x)g(x) dx }

Also written as:

∫udv=uv−∫vdu \int u dv = uv - \int v du

where

u=f(x)  ,  dv=g(x)dxdu=f′(x)  ,  v=∫g(x)dx \begin{aligned} u &= f(x) ~~,~~ dv = g(x)dx \\\\ du &=f'(x) ~~,~~ v = \int g(x) dx \end{aligned}

Exercise 16:

Integration Exercise 9

Try yourself
∫xsin⁡x dx= \color{blue}{\int x \sin x \,dx = }
sin⁡x+xcos⁡x+C \sin x + x\cos x + C cos⁡x+xsin⁡x+C \cos x + x\sin x + C
sin⁡x−xcos⁡x+C \sin x - x\cos x + C cos⁡x−xsin⁡x+C \cos x - x\sin x + C

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