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Topics You Need To Understand For This Page
Trig Identities and Formulae  NEW
basic trig identities 

\(\sin^2\theta+\cos^2\theta=1\)  \(1+\tan^2\theta=\sec^2\theta\) 
\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\)  \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\) 
\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\)  \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\) 
power reduction (halfangle) formulae 
\(\displaystyle{\sin^2\theta=\frac{1\cos(2\theta)}{2}}\)  \(\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}\) 
double angle formulae 
\(\sin(2\theta)=2\sin\theta\cos\theta\)  \(\cos(2\theta)=\cos^2\theta\sin^2\theta\) 
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Trig Derivatives and Integrals  NEW
basic trig derivatives  

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) 
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = \sin(t) }\)  
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) 
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = \csc^2(t) }\)  
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) 
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = \csc(t)\cot(t) }\)  
trig integrals  
\(\int{\sin(x)~dx} = \cos(x)+C\)  
\(\int{\cos(x)~dx} = \sin(x)+C\)  
\(\int{\tan(x)~dx} = \ln\abs{\cos(x)}+C\)  
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)  
\(\int{\sec(x)~dx} = \ln\abs{\sec(x)+\tan(x)}+C\)  
\(\int{\csc(x)~dx} = \ln\abs{\csc(x)+\cot(x)}+C\)  
reduction formulae  
Reduction Formulas (where n is a positive integer)  
\(\displaystyle{\int{\sin^n x~dx} = \frac{\sin^{n1}x\cos x}{n}+\frac{n1}{n}\int{\sin^{n2}x~dx}}\)  
\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n1}x\sin x}{n} + \frac{n1}{n}\int{\cos^{n2}x~dx}}\)  
Reduction Formulas (where n is an integer and \(n>1\))  
\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n1}x}{n1}  \int{\tan^{n2}x~dx}}\)  
\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n2}x\tan x}{n1}+\frac{n2}{n1}\int{\sec^{n2}x~dx}}\)  
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Trig Integration Practice Problems 

This page consists solely of practice problems using trig integration. The techniques used are basic trig integration, sinecosine trig integration and secanttangent trig integration. 
Instructions   Unless otherwise instructed, evaluate the following integrals. Give all answers in exact, simplified form. 
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basic techniques  
basic trig, algebra, substitution(17)  
special techniques & substitutions(2)  
advanced techniques  
sinecosine integrals(23)  
secanttangent integrals(11) 
Level A  Basic 
Practice A01  

\(\displaystyle{\int{a\cos(x)+\frac{b}{\sin^2(x)}dx}}\)  
solution 
Practice A03  

\(\displaystyle{\int{\sin(4x)\cos(2x)~dx}}\)  
solution 
Practice A04  

\(\displaystyle{\int{\cos^5x\sin^5x~dx}}\)  
solution 
Practice A05  

\(\displaystyle{\int{\cos^4x\sin^3x~dx}}\)  
solution 
Practice A07  

\(\displaystyle{\int{\sin^3(x)\cos(x)~dx}}\)  
solution 
Practice A08  

\(\displaystyle{\int{2\sin(x)\cos(x)~dx}}\)  
solution 
Practice A09  

\(\displaystyle{\int{\sec(x)\tan(x)~dx}}\)  
solution 
Practice A10  

\(\displaystyle{\int{\sin^3x~\cos^2x~dx}}\)  
solution 
Practice A11  

\(\displaystyle{\int{\sin^3x~\cos^3x~dx}}\)  
solution 
Practice A12  

\(\displaystyle{\int{\sin^2x~\cos^2x~dx}}\)  
solution 
Practice A13  

\(\displaystyle{\int{\sec^4x~\tan^2x~dx}}\)  
solution 
Practice A14  

\(\displaystyle{\int{\sec^4x~dx}}\)  
solution 
Practice A15  

\(\displaystyle{\int{\sec^3x~\tan^5x~dx}}\)  
solution 
Practice A16  

\(\displaystyle{\int{\tan(x)~dx}}\)  
solution 
Practice A17  

\(\displaystyle{\int{\sin(2x)~\cos(3x)~dx}}\)  
solution 
Practice A18  

\(\displaystyle{\int{\sin(2x)~\sin(3x)~dx}}\)  
solution 
Practice A19  

\(\displaystyle{\int{\frac{\cos(x)+\sin(x)}{\sin(2x)}~dx}}\)  
solution 
Practice A20  

\(\displaystyle{\int{\frac{\sqrt[3]{\cot(x)}}{\sin^2(x)}~dx}}\)  
solution 
Practice A21  

\(\displaystyle{\int{\sec^4x~\tan^6x~dx}}\)  
solution 
Practice A22  

\(\displaystyle{\int{\frac{1}{\cos^6(x)\cot^2(x)}~dx}}\)  
solution 
Practice A23  

\(\displaystyle{\int{\frac{\cos^5(x)\sin(x)}{1\sin^2(x)}dx}}\)  
solution 
Practice A24  

\(\displaystyle{\int{\tan^3(x)(\csc^2(x)1)~dx}}\)  
solution 
Practice A25  

\(\displaystyle{\int{\sin^3x~\sec^2x~dx}}\)  
solution 
Practice A26  

\(\displaystyle{\int{\sin x~\cos(2x)~dx}}\)  
solution 
Practice A27  

\(\displaystyle{\int{\sin(2x)\cos(3x)~dx}}\)  
solution 
Practice A28  

\(\displaystyle{\int_{0}^{\pi/2}{\sin^2x~\cos^2x~dx}}\)  
solution 
Practice A29  

\(\displaystyle{\int{\tan^3(x)~\sec(x)~dx}}\)  
solution 
Practice A30  

\(\displaystyle{\int{\tan^4x~\sec^6x~dx}}\)  
solution 
Practice A31  

\(\displaystyle{\int{\sin(8x)~\cos(5x)~dx}}\)  
solution 
Practice A32  

\(\displaystyle{\int{\sin(3x)\sin(6x)~dx}}\)  
solution 
Practice A33  

\(\displaystyle{\int{\cos(4\pi x)\cos(\pi x)~dx}}\)  
solution 
Practice A34  

\(\displaystyle{\int{3(x^5)\sin(x^6)~dx}}\)  
solution 
Practice A35  

\(\displaystyle{\int{\frac{\sin\sqrt{x}}{\sqrt{x}}~dx}}\)  
solution 
Practice A36  

\(\displaystyle{\int{5\cos^4(2x)\sin(2x)~dx}}\)  
solution 
Practice A37  

\(\displaystyle{ \int_{\pi/2}^{\pi}{ \sin^3 \theta ~ \cos^2 \theta ~ d\theta} }\)  
solution 
Level B  Intermediate 
Practice B03  

\(\displaystyle{\int_{0}^{\pi/2}{x\cos(x)~dx}}\)  
solution 
Practice B04  

\(\displaystyle{\int_{0}^{\pi/2}{\sin^7\theta~\cos^5\theta~d\theta}}\)  
solution 
Practice B05  

\(\displaystyle{\int{\cos^4x~dx}}\)  
solution 
Practice B06  

\(\displaystyle{\int{\tan^5x~dx}}\)  
solution 
Practice B07  

\(\displaystyle{\int{\cos^2x~\tan^3x~dx}}\)  
solution 
Practice B08  

\(\displaystyle{\int{\csc(x)~dx}}\)  
solution 
Practice B09  

\(\displaystyle{\int{\frac{\cos(x)\sin(\csc~x)}{\sin^2(x)}dx}}\)  
solution 
Practice B10  

\(\displaystyle{\int{\sin^2(\pi x)~\cos^5(\pi x)~dx}}\)  
solution 
Practice B11  

\(\displaystyle{\int{\sec(x)~dx}}\)  
solution 
Practice B12  

\(\displaystyle{\int{\sqrt{x}\sec(x^{3/2})\tan(x^{3/2})~dx}}\)  
solution 
Practice B13  

\(\displaystyle{\int{(1+2t^2)^2~t~\csc^2\left[(1+2t^2)^3\right]~dt}}\)  
solution 
Level C  Advanced 
Practice C02  

\(\displaystyle{\int_{0}^{\pi/2}{\sin^7x~dx}}\)  
solution 