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17calculus > integrals > trig integration > practice problems

### 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

$$\sin(2\theta)=2\sin\theta\cos\theta$$   |   $$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

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 $$n>1$$)

$$\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx}}$$

$$\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}\int{\sec^{n-2}x~dx}}$$

17calculus trig formulas - full list

### Calculus Main Topics

Integrals

Integral Applications

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

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free ideas to save on books - bags - supplies 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.

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.

### Search 17Calculus

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basic techniques

basic trig, algebra, substitution(17)

special techniques & substitutions(2)

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

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

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

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

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

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

solution

 Level B - Intermediate

Practice B01

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

solution

Practice B02

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

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

Practice C01

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

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