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17calculus > integrals > trig integration > practice problems |
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 |
ATTENTION INSTRUCTORS: The new 2018 version of 17calculus will include changes to the practice problem numbering system. If you would like advance information to help you prepare for spring semester, send us an email at 2018info at 17calculus.com. |
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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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\(\displaystyle{\int{a\cos(x)+\frac{b}{\sin^2(x)}dx}}\) | |
solution |
Practice A03 | |
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\(\displaystyle{\int{\sin(4x)\cos(2x)~dx}}\) | |
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Practice A04 | |
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\(\displaystyle{\int{\cos^5x\sin^5x~dx}}\) | |
solution |
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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\(\displaystyle{\int{\sec(x)\tan(x)~dx}}\) | |
solution |
Practice A10 | |
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\(\displaystyle{\int{\sin^3x~\cos^2x~dx}}\) | |
solution |
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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\(\displaystyle{\int{\tan(x)~dx}}\) | |
solution |
Practice A17 | |
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\(\displaystyle{\int{\sin(2x)~\cos(3x)~dx}}\) | |
solution |
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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\(\displaystyle{\int{\sec^4x~\tan^6x~dx}}\) | |
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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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\(\displaystyle{\int{\tan^3(x)(\csc^2(x)-1)~dx}}\) | |
solution |
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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\(\displaystyle{\int{\sin(8x)~\cos(5x)~dx}}\) | |
solution |
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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\(\displaystyle{\int{\frac{\sin\sqrt{x}}{\sqrt{x}}~dx}}\) | |
solution |
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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\(\displaystyle{\int{\csc(x)~dx}}\) | |
solution |
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 |