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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

\(\sin^2\theta+\cos^2\theta=1\)   |   \(1+\tan^2\theta=\sec^2\theta\)

\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\)   |   \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\)

\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\)   |   \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\)

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

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

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)

\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n}+\frac{n-1}{n}\int{\sin^{n-2}x~dx}}\)

\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n-1}x\sin x}{n} + \frac{n-1}{n}\int{\cos^{n-2}x~dx}}\)

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

17calculus trig formulas - full list

Calculus Main Topics

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Related Topics and Links

Trig Integration Practice Problems

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.

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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

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

solution

Practice A02

\(\displaystyle{\int{\sin^4(2x)\cos(2x)~dx}}\)

answer

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 A06

\(\displaystyle{\int{x\cos(x^2+1)~dx}}\)

answer

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

Practice A38

\(\int{\cos^5(\theta)~d\theta}\)

answer

solution


Level B - Intermediate

Practice B01

\(\displaystyle{\int{\frac{dx}{2\sin(x)+\sin(2x)}}}\)

answer

solution

Practice B02

\(\displaystyle{\int{\cos^4x~\sin^2x~dx}}\)

answer

solution

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 C01

\(\displaystyle{\int_{\pi/2}^{3\pi/2}{\sqrt{1+\sin\theta}~d\theta}}\)

answer

solution

Practice C02

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

solution

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