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Wize University Calculus 1 Textbook > Integrals

Antiderivatives of Trig Functions

Antiderivatives & The Indefinite IntegralsPrevious Section
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Antiderivatives (Indefinite Integral) of Trig Functions

Let kk be a nonzero constant. Then we have the following indefinite integral rules:
∫sin⁡(kx)dx=−1kcos⁡(kx)+C∫cos⁡(kx)dx=1ksin⁡(kx)+C∫sec⁡2(kx)dx=1ktan⁡(kx)+C∫sec⁡(kx)tan⁡(kx)dx=1ksec⁡(kx)+C∫csc⁡2(kx)dx=−1kcot⁡(kx)+C∫csc⁡(kx)cot⁡(kx)dx=−1kcsc⁡(kx)+C\begin{array}{|l|l|} \hline\\ \displaystyle\int\sin{(kx)}dx=-\dfrac{1}{k}\cos{( kx)}+C & \displaystyle\int\cos{ (kx)}dx=\dfrac{1}{k}\sin{( kx)}+C \\ \\ \hline\\ \displaystyle\int\sec^2{(kx)}dx=\dfrac{1}{k}\tan (kx)+C &\displaystyle\int\sec (kx)\tan (kx)dx=\dfrac{1}{k}\sec (kx)+C \\ \\ \hline\\ \displaystyle\int\csc^2(kx)dx=-\dfrac{1}{k}\cot (kx)+C & \displaystyle\int\csc (kx)\cot (kx)dx=-\dfrac{1}{k}\csc( kx)+C \\ \\ \hline \end{array}
Additionally
∫tan⁡xdx=−ln⁡∣cos⁡x∣+C∫cot⁡xdx=ln⁡∣sin⁡x∣∫sec⁡xdx=ln⁡∣sec⁡x+tan⁡x∣+C∫csc⁡xdx=−ln⁡∣csc⁡x+cot⁡x∣+C\begin{array}{|l|l|} \hline\\ \displaystyle \int\tan xdx=-\ln|\cos x|+C & \displaystyle \int \cot xdx=\ln|\sin x| \\ \\ \hline\\ \displaystyle\int \sec x dx=\ln|\sec x+\tan x| + C &\displaystyle \int \csc x dx=-\ln|\csc x+\cot x|+C \\ \\ \hline \end{array}

Wize Tip
These last four indefinite integrals can be found using future techniques of integration and are not always required to be memorized.

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Example: Trig Indefinite Integrals

Evaluate the following indefinite integral

∫1+tan⁡2θ dθ\displaystyle \int\sqrt{1+\tan^2\theta} \ d\theta

∫1+tan⁡2θ dθ\displaystyle \int\sqrt{1+\tan^2\theta} \ d\theta

=∫sec⁡2θ dθ=\displaystyle \int\sqrt{\sec^2\theta} \ d\theta

=∫sec⁡θ dθ=\displaystyle \int\sec \theta \ d\theta

=ln⁡∣sec⁡θ+tan⁡θ∣+C\displaystyle=\ln|\sec \theta+\tan \theta| + C
Evaluate the following indefinite integral

∫(x2+2sin⁡x−3cos⁡x)dx\displaystyle \int( x^2+2\sin x-3\cos x )dx
Evaluate the following indefinite integral

∫(sec⁡2x)(cos⁡22x)dx\displaystyle \int (\sec 2x)( \cos^2 2x)dx

Extra Practice
Derivatives/Antiderivatives
What is the derivative of f(x)=excos⁡xf(x)=e^{x}\cos x?
Derivatives/Antiderivatives
What is the derivative of f(x)=e2xcos⁡3xf(x)=e^{2x}\cos3x?
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Practice: Indefinite Integral
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Antiderivatives: Trigonometric Functions
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Find the value of the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\displaystyle \int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx .
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Practice: Indefinite Integral (~F2018 Final Q26)
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
Indefinite Integrals with Trig and Inverse Trig
Q:\textbf{Q:} Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Antiderivatives: Indefinite with Trig
Find ∫(tan⁡2x+1sec⁡2x+1+xπ+sin⁡2x)dx\displaystyle\int_{ }^{ }\left(\tan^2x+\frac{1}{\sec^2x}+1+x^\pi+\sin^2x\right)dx
Antiderivatives: Product with Trig
Find dfdθ\displaystyle\frac{df}{d\theta} given f(θ)=(sin⁡θ−cos⁡θ)sec⁡θ\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta
Practice: Indefinite Integral
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Practice: Indefinite Integral
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Antiderivatives: Trigonometric Functions
Find
∫1+sin⁡3xsin⁡2xdx\int \frac{1+\sin^3 x}{\sin^2 x} dx
Find an antiderivative for
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2
Antiderivatives: Trigonometric Functions
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
final114
Find the indefinite integral ∫(1x2+sin⁡x)dx\displaystyle\int\bigg(\frac{1}{x^2}+\sin x\bigg)dx
Antiderivatives: Trigonometric Functions
Evaluate the integral∫[tan⁡2x+1+1sec⁡2x+sin⁡2x]dx\displaystyle \int_{ }^{ }\left[\tan^2x+1+\frac{1}{\sec^2x}+\sin^2x\right]dx.
Hint: First simplify using trig identities
final114
Find the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx
Find the exact value of
∫0πsin⁡xdx\int_0^{\pi}\sin xdx
Antiderivatives: Product with Trig
Find dfdθ\displaystyle\frac{df}{d\theta} given f(θ)=(sin⁡θ−cos⁡θ)sec⁡θ\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta
Find the indefinite integral ∫(1x2+sin⁡x)dx\displaystyle\int\bigg(\frac{1}{x^2}+\sin x\bigg)dx . Use C for the integration constant.
Antiderivatives: Indefinite with Trig
Find ∫(tan⁡2x+1sec⁡2x+1+xπ+sin⁡2x)dx\displaystyle\int_{ }^{ }\left(\tan^2x+\frac{1}{\sec^2x}+1+x^\pi+\sin^2x\right)dx
Indefinite Integrals with Trig and Inverse Trig
Find ∫(2x−34−x2+2sin⁡x)dx\displaystyle\int_{ }^{ }\left(2\sqrt{x}-\frac{3}{\sqrt{4-x^2}}+2\sin x\right)dx
Derivatives/Antiderivatives
Compute the second derivative of f(x)=etan⁡xf(x)=e^{\tan x}.
Find the value of the definite integral ∫π/6π/3(sec⁡x+tan⁡x)sec⁡xdx\displaystyle \int_{\pi/6 }^{\pi/3}(\sec x+\tan x)\sec x dx .
Evaluate the indefinite integral ∫sec⁡θtan⁡θ−sin⁡(θ5)dθ\displaystyle \int\sec\theta\tan\theta-\sin\left(\frac{\theta}{5}\right)d\theta.
Find ∫0π∣cos⁡ x∣dx\displaystyle\int_0^{\pi}\left|\cos\ x\right|dx
Antiderivatives: Trigonometric Functions
Evaluate ∫π3π∣cos⁡x∣dx\int_{\frac{\pi}{3}}^{\pi}\left|\cos x\right|dx.
Antiderivatives: Trigonometric and Exponential Functions
If f′′(x)=cos⁡(2x)−x4+exf''\left(x\right)=\cos\left(2x\right)-x^4+e^x, f(0)=0=f′(0)f\left(0\right)=0=f'\left(0\right)
Practice: Indefinite Integral
Practice: Indefinite Integral
Find f(x)f\left(x\right), given that f′(x)=x2−2+sin⁡xf'\left(x\right)=x^2-2+\sin x and f(0)=3f\left(0\right)=3.
For each of the following functions, find the most general antiderivative.
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2
Find the most general antiderivative.
f(x)=−cos⁡x+x2f(x)=-\cos{x}+x^2