Marcus is designing a skatepark ramp. The slope of the ramp at horizontal distance $x$ (in feet) is $\sin^2 x$, and the ramp starts at height $0$. How tall is the ramp at any $x$? You need an antiderivative of $\sin^2 x$, and it is not in your table of basic antiderivatives. The good news: there is no brand-new idea on this page. It is your basic table, the chain-rule pattern, and trig identities, plus integration by parts for the toughest cases.
Basic trig antiderivatives and linear arguments
Every derivative you know run backward gives an antiderivative. Here are the trig ones to have memorized.
Some problems call an antiderivative a primitive. It is the same thing: a function $F$ with $F'=f$.
What if the argument is not just $x$? If $F'=f$, then by the chain rule $\dfrac{d}{dx}F(ax+b)=a\,f(ax+b)$. So to undo it, divide by $a$.
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To start practicing calculating antiderivatives, find the general primitive of the following function which includes a linear adjustment in the argument of the cosine:…
To start practicing calculating antiderivatives, find the general primitive of the following function which includes a linear adjustment in the argument of the cosine:
Find the antiderivative of the function $f(x) = \sec^2(3x)$ given that its graph passes through the point $P(0, 2)$. Remember that you must calculate the constant of…
Find the antiderivative of the function $f(x) = \sec^2(3x)$ given that its graph passes through the point $P(0, 2)$. Remember that you must calculate the constant of integration $C$.
A simple harmonic oscillator moves in such a way that its velocity as a function of time $t$ (in seconds) is given by $v(t) = 3 \cdot \cos(t)$. If at the initial instant…
A simple harmonic oscillator moves in such a way that its velocity as a function of time $t$ (in seconds) is given by $v(t) = 3 \cdot \cos(t)$. If at the initial instant $t=0$ the object is at the position $s=0$, find the expression for its position $s(t)$ by calculating the corresponding antiderivative.
Calculate the following indefinite integral using the natural logarithm as the result, recognizing that the numerator is (almost) the derivative of the denominator:…
Calculate the following indefinite integral using the natural logarithm as the result, recognizing that the numerator is (almost) the derivative of the denominator:
Determine the family of primitive functions of the given expression. Bear in mind that the derivative of sine is cosine: $f(x) = \sin^4(x) \cdot \cos(x)$
Determine the family of primitive functions of the given expression. Bear in mind that the derivative of sine is cosine:
Find the indefinite integral of the function $f(x) = \tan^3(x) \cdot \sec^2(x)$. Remember to observe if the expression fits a power-type integral where a function is…
Find the indefinite integral of the function $f(x) = \tan^3(x) \cdot \sec^2(x)$. Remember to observe if the expression fits a power-type integral where a function is raised to an exponent and multiplied by its derivative.
Calculate the primitive $F(x)$ of the function $f(x) = \frac{\sin(x)}{\cos^3(x)}$ that satisfies the condition of passing through the origin of coordinates, that is,…
Calculate the primitive $F(x)$ of the function $f(x) = \frac{\sin(x)}{\cos^3(x)}$ that satisfies the condition of passing through the origin of coordinates, that is, $F(0) = 0$.
Hint: Identify whether it is a logarithmic or power-type integral before applying the initial conditions.
Use them to turn a "mystery" expression into a sum of terms from the basic table.
The two curves are the same: $\sec^2x-1=\tan^2x$.
Powers of $\csc x$ and $\cot x$
A power of cosecant works like a power of secant. Peel off $\csc^2x$, which is $-1$ times the derivative of $\cot x$, and turn the rest into cotangents with $\csc^2x=1+\cot^2x$.
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On many occasions, trigonometric identities allow us to transform a difficult expression into a standard one. Use the relationship $\tan^2(x) + 1 = \sec^2(x)$ to…
On many occasions, trigonometric identities allow us to transform a difficult expression into a standard one. Use the relationship $\tan^2(x) + 1 = \sec^2(x)$ to calculate the following primitive:
Determine the indefinite integral of the function $f(x) = \frac{1}{\sin^2(x) \cdot \cos^2(x)}$. Help: You can try substituting the number $1$ in the numerator with the…
Determine the indefinite integral of the function $f(x) = \frac{1}{\sin^2(x) \cdot \cos^2(x)}$.
Help: You can try substituting the number $1$ in the numerator with the fundamental identity $\sin^2(x) + \cos^2(x) = 1$ to separate the integral into two simpler terms.
Calculate the primitive of the following trigonometric function using fundamental identities to simplify the expression before integrating: $$f(x) = \frac{\cos^3(x)}{1 -…
Calculate the primitive of the following trigonometric function using fundamental identities to simplify the expression before integrating:
Determine the primitive of the function $f(x)$ given that its graph passes through the point $P(\pi, 1)$ and its derivative is: $$f'(x) = \frac{1}{\sin^4(x) \cdot…
Determine the primitive of the function $f(x)$ given that its graph passes through the point $P(\pi, 1)$ and its derivative is:
$$f'(x) = \frac{1}{\sin^4(x) \cdot \cos^2(x)}$$
Hint: Try to express the numerator using the fundamental identity $\sin^2(x) + \cos^2(x) = 1$ iteratively or look for a substitution based on the tangent.
When $\sin x$ or $\cos x$ appears to an odd power, peel off one factor. That factor is $du$ for a substitution, and the rest converts with $\sin^2x+\cos^2x=1$.
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Calculate the primitive of the function $f(x) = \frac{\sin^3(x)}{\cos^2(x)}$. Hint: As it is an odd function in $\sin(x)$, use the substitution $t = \cos(x)$.
Calculate the primitive of the function $f(x) = \frac{\sin^3(x)}{\cos^2(x)}$.
Hint: As it is an odd function in $\sin(x)$, use the substitution $t = \cos(x)$.
Calculate the indefinite integral of the following rational trigonometric function. These types of integrals frequently appear in the calculation of gravitational fields…
Calculate the indefinite integral of the following rational trigonometric function. These types of integrals frequently appear in the calculation of gravitational fields in astrophysics:
An architecture firm is designing the profile of a ramp for an extreme sports park. The gradient of the ramp at each point $x$ is given by the function…
An architecture firm is designing the profile of a ramp for an extreme sports park. The gradient of the ramp at each point $x$ is given by the function $g(x) = \sin^2(x)$. If the ramp starts at the point $(0, 0)$, determine the function $G(x)$ that describes the height of the ramp at any point.
Hint: Use the double angle formula to reduce the degree of the function.
Products of sines and cosines with different angles
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Find the primitive of the function $f(x)$ described below, using product-to-sum transformations: $$f(x) = \cos(5x) \cdot \cos(3x)$$ Once the family of primitives is…
Find the primitive of the function $f(x)$ described below, using product-to-sum transformations:
$$f(x) = \cos(5x) \cdot \cos(3x)$$
Once the family of primitives is obtained, determine the one whose graph has a relative maximum at the point with x-coordinate $x = \frac{\pi}{2}$ and y-coordinate $y = 2$.
Some integrals mix a trig function with a polynomial or an exponential. Identities will not help. Use the product rule backward.
The second parts step above is what the table method automates for a polynomial times $\sin$, $\cos$, or $e^x$: differentiate down one column, integrate down the other, and multiply along the diagonals with alternating signs.
Another trap: some integrals come back to themselves. They are called cyclic.
For a mix such as $e^{2x}\cos^2x$, first use the double angle formula to remove the square, then handle the cyclic piece $e^{2x}\cos 2x$.
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Find the primitive of the function $h(x) = x^2 \cdot \cos(x)$ using the method of integration by parts. You will need to apply the method twice to reduce the degree of…
Find the primitive of the function $h(x) = x^2 \cdot \cos(x)$ using the method of integration by parts. You will need to apply the method twice to reduce the degree of the polynomial until you obtain a standard integral.
In a physics experiment, the damped oscillation of a spring is analyzed, whose vibration velocity as a function of time $t$ behaves according to the function…
In a physics experiment, the damped oscillation of a spring is analyzed, whose vibration velocity as a function of time $t$ behaves according to the function $v(t) = e^{-t} \cdot \sin(2t)$. Determine the position function $s(t)$ given that this integral is cyclic.
In the design of alternating current circuits, instantaneous power is usually given by the product of exponential and trigonometric functions. Calculate the family of…
In the design of alternating current circuits, instantaneous power is usually given by the product of exponential and trigonometric functions. Calculate the family of primitive functions of the following expression, which represents the rate of change of energy in a specific component:
$$f(x) = e^{2x} \cdot \cos^2(x)$$
Tip: First use a trigonometric identity to reduce the exponent of the cosine before applying the method of integration by parts.
Two more standard forms help when a square root or a sum of squares shows up.
Integrating an inverse trig function by parts
You cannot integrate $\arcsin$ from the table, but it has a simple derivative. That makes it a good choice for $u$ in integration by parts, with $dv=dx$.
Even powers in a fraction
Rational functions of $\sin^2x$ and $\cos^2x$ (only even powers) are handled by dividing the numerator and the denominator by $\cos^2x$. That makes $\sec^2x$, the derivative of $\tan x$, appear.
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Calculate the indefinite integral of the function $f(x)$, which combines powers of inverse trigonometric functions and direct functions: $$I = \int \frac{\arcsen(x) +…
Calculate the indefinite integral of the function $f(x)$, which combines powers of inverse trigonometric functions and direct functions:
Hint: Decompose the integral into a sum of two simpler integrals and analyze whether either of them can be solved as an integral of the type "function multiplied by its derivative".
Find the family of primitives of the following function using a suitable substitution for even trigonometric functions in sine and cosine: $$f(x) = \frac{1}{3\sin^2(x) +…
Find the family of primitives of the following function using a suitable substitution for even trigonometric functions in sine and cosine:
$$f(x) = \frac{1}{3\sin^2(x) + \cos^2(x)}$$
Hint: Divide the numerator and denominator by $\cos^2(x)$ to express the function in terms of $\tan(x)$ and its derivative $\sec^2(x)$.
In the study of the oscillations of a suspension bridge, the acceleration of a point on the structure is given by the function $a(t)$. Find the position function $s(t)$…
In the study of the oscillations of a suspension bridge, the acceleration of a point on the structure is given by the function $a(t)$. Find the position function $s(t)$ given that the initial velocity is $v(0) = 0$, the initial position is $s(0) = 1$ and the acceleration is:
$$a(t) = \frac{\sec^2(t)}{\sqrt{4 - \tan^2(t)}}$$
Help: You will need to integrate twice. The first integral will give you the velocity and the second the position. Be careful with the domains of the resulting functions.
When nothing else works on a rational expression in $\sin x$ and $\cos x$, a last resort is the so-called universal substitution $t=\tan\frac x2$. It turns the integral into a rational function of $t$.
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The "universal substitution" is a powerful tool for solving integrals of rational functions of sine and cosine. Use the substitution $t = \tan\left(\frac{x}{2}\right)$…
The "universal substitution" is a powerful tool for solving integrals of rational functions of sine and cosine. Use the substitution $t = \tan\left(\frac{x}{2}\right)$ to find the following primitive: