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Topics You Need To Understand For This Page
Trig Identities and Formulae - NEW
basic trig identities |
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\(\sin^2\theta+\cos^2\theta=1\) | \(1+\tan^2\theta=\sec^2\theta\) |
\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\) | \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\) |
\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\) | \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\) |
power reduction (half-angle) formulae |
\(\displaystyle{\sin^2\theta=\frac{1-\cos(2\theta)}{2}}\) | \(\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}\) |
double angle formulae |
\(\sin(2\theta)=2\sin\theta\cos\theta\) | \(\cos(2\theta)=\cos^2\theta-\sin^2\theta\) |
links |
Trig Derivatives and Integrals - NEW
basic trig derivatives | ||
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\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) | |
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) | ||
\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n}+\frac{n-1}{n}\int{\sin^{n-2}x~dx}}\) | ||
\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n-1}x\sin x}{n} + \frac{n-1}{n}\int{\cos^{n-2}x~dx}}\) | ||
Reduction Formulas (where n is an integer and \(n>1\)) | ||
\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx}}\) | ||
\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}\int{\sec^{n-2}x~dx}}\) | ||
links | ||
Calculus Main Topics
Integrals |
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Integral Applications |
Single Variable Calculus |
Multi-Variable Calculus |
Tools
math tools |
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general learning tools |
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Related Topics and Links
related topics on other pages |
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external links you may find helpful |
Trig Integration Practice Problems |
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This page consists solely of practice problems using trig integration. The techniques used are basic trig integration, sine-cosine trig integration and secant-tangent trig integration. |
Instructions - - Unless otherwise instructed, evaluate the following integrals. Give all answers in exact, simplified form. |
Search 17Calculus |
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basic techniques | |
basic trig, algebra, substitution(17) | |
special techniques & substitutions(2) | |
advanced techniques | |
sine-cosine integrals(23) | |
secant-tangent integrals(11) |
Level A - Basic |
Practice A01 | |
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\(\displaystyle{\int{a\cos(x)+\frac{b}{\sin^2(x)}dx}}\) | |
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Practice A03 | |
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\(\displaystyle{\int{\sin(4x)\cos(2x)~dx}}\) | |
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Practice A04 | |
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\(\displaystyle{\int{\cos^5x\sin^5x~dx}}\) | |
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Practice A05 | |
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\(\displaystyle{\int{\cos^4x\sin^3x~dx}}\) | |
solution |
Practice A07 | |
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\(\displaystyle{\int{\sin^3(x)\cos(x)~dx}}\) | |
solution |
Practice A08 | |
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\(\displaystyle{\int{2\sin(x)\cos(x)~dx}}\) | |
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Practice A09 | |
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\(\displaystyle{\int{\sec(x)\tan(x)~dx}}\) | |
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Practice A10 | |
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\(\displaystyle{\int{\sin^3x~\cos^2x~dx}}\) | |
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Practice A11 | |
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\(\displaystyle{\int{\sin^3x~\cos^3x~dx}}\) | |
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Practice A12 | |
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\(\displaystyle{\int{\sin^2x~\cos^2x~dx}}\) | |
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Practice A13 | |
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\(\displaystyle{\int{\sec^4x~\tan^2x~dx}}\) | |
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Practice A14 | |
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\(\displaystyle{\int{\sec^4x~dx}}\) | |
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Practice A15 | |
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\(\displaystyle{\int{\sec^3x~\tan^5x~dx}}\) | |
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Practice A16 | |
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\(\displaystyle{\int{\tan(x)~dx}}\) | |
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Practice A17 | |
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\(\displaystyle{\int{\sin(2x)~\cos(3x)~dx}}\) | |
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Practice A18 | |
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\(\displaystyle{\int{\sin(2x)~\sin(3x)~dx}}\) | |
solution |
Practice A19 | |
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\(\displaystyle{\int{\frac{\cos(x)+\sin(x)}{\sin(2x)}~dx}}\) | |
solution |
Practice A20 | |
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\(\displaystyle{\int{\frac{\sqrt[3]{\cot(x)}}{\sin^2(x)}~dx}}\) | |
solution |
Practice A21 | |
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\(\displaystyle{\int{\sec^4x~\tan^6x~dx}}\) | |
solution |
Practice A22 | |
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\(\displaystyle{\int{\frac{1}{\cos^6(x)\cot^2(x)}~dx}}\) | |
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Practice A23 | |
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\(\displaystyle{\int{\frac{\cos^5(x)\sin(x)}{1-\sin^2(x)}dx}}\) | |
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Practice A24 | |
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\(\displaystyle{\int{\tan^3(x)(\csc^2(x)-1)~dx}}\) | |
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Practice A25 | |
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\(\displaystyle{\int{\sin^3x~\sec^2x~dx}}\) | |
solution |
Practice A26 | |
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\(\displaystyle{\int{\sin x~\cos(2x)~dx}}\) | |
solution |
Practice A27 | |
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\(\displaystyle{\int{\sin(2x)\cos(3x)~dx}}\) | |
solution |
Practice A28 | |
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\(\displaystyle{\int_{0}^{\pi/2}{\sin^2x~\cos^2x~dx}}\) | |
solution |
Practice A29 | |
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\(\displaystyle{\int{\tan^3(x)~\sec(x)~dx}}\) | |
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Practice A30 | |
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\(\displaystyle{\int{\tan^4x~\sec^6x~dx}}\) | |
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Practice A31 | |
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\(\displaystyle{\int{\sin(8x)~\cos(5x)~dx}}\) | |
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Practice A32 | |
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\(\displaystyle{\int{\sin(3x)\sin(6x)~dx}}\) | |
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Practice A33 | |
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\(\displaystyle{\int{\cos(4\pi x)\cos(\pi x)~dx}}\) | |
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Practice A34 | |
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\(\displaystyle{\int{3(x^5)\sin(x^6)~dx}}\) | |
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Practice A35 | |
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\(\displaystyle{\int{\frac{\sin\sqrt{x}}{\sqrt{x}}~dx}}\) | |
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Practice A36 | |
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\(\displaystyle{\int{5\cos^4(2x)\sin(2x)~dx}}\) | |
solution |
Practice A37 | |
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\(\displaystyle{ \int_{\pi/2}^{\pi}{ \sin^3 \theta ~ \cos^2 \theta ~ d\theta} }\) | |
solution |
Level B - Intermediate |
Practice B03 | |
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\(\displaystyle{\int_{0}^{\pi/2}{x\cos(x)~dx}}\) | |
solution |
Practice B04 | |
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\(\displaystyle{\int_{0}^{\pi/2}{\sin^7\theta~\cos^5\theta~d\theta}}\) | |
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Practice B05 | |
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\(\displaystyle{\int{\cos^4x~dx}}\) | |
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Practice B06 | |
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\(\displaystyle{\int{\tan^5x~dx}}\) | |
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Practice B07 | |
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\(\displaystyle{\int{\cos^2x~\tan^3x~dx}}\) | |
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Practice B08 | |
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\(\displaystyle{\int{\csc(x)~dx}}\) | |
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Practice B09 | |
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\(\displaystyle{\int{\frac{\cos(x)\sin(\csc~x)}{\sin^2(x)}dx}}\) | |
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Practice B10 | |
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\(\displaystyle{\int{\sin^2(\pi x)~\cos^5(\pi x)~dx}}\) | |
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Practice B11 | |
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\(\displaystyle{\int{\sec(x)~dx}}\) | |
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Practice B12 | |
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\(\displaystyle{\int{\sqrt{x}\sec(x^{3/2})\tan(x^{3/2})~dx}}\) | |
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Practice B13 | |
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\(\displaystyle{\int{(1+2t^2)^2~t~\csc^2\left[(1+2t^2)^3\right]~dt}}\) | |
solution |
Level C - Advanced |
Practice C02 | |
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\(\displaystyle{\int_{0}^{\pi/2}{\sin^7x~dx}}\) | |
solution |