The indefinite integral. Elementary decomposition and standard integrals
You know a function $f$ and you are given its derivative, $f'$. Until now you have always gone forwards: from $f$ to $f'$. Now the problem is reversed: you are given $f'$ and you have to recover $f$. It is exactly the same table of derivatives from 1st and 2nd Bachillerato, but read in the opposite direction.
Differentiating backwards: what an antiderivative is
Think of a car: if $s(t)$ is the position and $v(t)=s'(t)$ the velocity, knowing $v(t)$ and wanting to recover $s(t)$ is exactly this problem. It is the opposite operation to differentiating.
Determine whether the function $F(x) = x \cdot e^x$ is an antiderivative of the function $f(x) = (x+1)e^x$. To justify your answer, remember to use the fundamental…
Determine whether the function $F(x) = x \cdot e^x$ is an antiderivative of the function $f(x) = (x+1)e^x$. To justify your answer, remember to use the fundamental relationship between a function and its antiderivative via differentiation.
Infinitely many antiderivatives: the constant $C$
To pick one particular antiderivative out of all of them, one extra piece of information is enough: the value of $F$ at a point.
Find the antiderivative of the function $f(x) = 2x - 6$ whose graph passes through the point $P(3, 4)$. Note: First calculate the indefinite integral to obtain the…
Find the antiderivative of the function $f(x) = 2x - 6$ whose graph passes through the point $P(3, 4)$.
Note: First calculate the indefinite integral to obtain the family of functions with the constant $C$, and then use the point $P$ to find the specific value of that constant.
A surveillance drone moves such that its horizontal velocity (in m/s) is given by the function $v(t) = 4e^{2t-1} + \sin(3t)$. Given that at the initial instant $t=0$ the…
A surveillance drone moves such that its horizontal velocity (in m/s) is given by the function $v(t) = 4e^{2t-1} + \sin(3t)$. Given that at the initial instant $t=0$ the drone is at position $s=0$, determine the expression for the position function $s(t)$.
The table of derivatives, read backwards
Each differentiation rule in the table from 1st and 2nd Bachillerato gives, read from right to left, a standard integral. There is nothing new to memorise: it is the same table.
Find the family of antiderivatives of the following function, simplifying the result by eliminating negative exponents and using radicals:…
Find the family of antiderivatives of the following function, simplifying the result by eliminating negative exponents and using radicals:
$f(x) = \dfrac{3}{x^4} - \dfrac{2}{\sqrt[3]{x^2}} + \dfrac{1}{5x}$
Find all antiderivatives of the function $f(x) = \frac{x^4 - 2x^2 + 5}{x^2}$.
Find all antiderivatives of the function $f(x) = \frac{x^4 - 2x^2 + 5}{x^2}$.
Linearity: sums, differences and constants
Calculate the indefinite integral of the following combination of trigonometric functions and constants, applying linearity properties (factoring out constants and…
Calculate the indefinite integral of the following combination of trigonometric functions and constants, applying linearity properties (factoring out constants and splitting sums):
$$\int (4 \sin x - 5 \cos x + 3) \, dx$$In the design of a smartphone camera lens, the curvature function needs to be known. The derivative of this function is $g'(x) = \frac{7}{1 + x^2} - 2^x$. Obtain the…
In the design of a smartphone camera lens, the curvature function needs to be known. The derivative of this function is $g'(x) = \frac{7}{1 + x^2} - 2^x$. Obtain the expression for the function $g(x)$ by evaluating its indefinite integral:
$$\int \left( \frac{7}{1+x^2} - 2^x \right) \, dx$$Find the indefinite integral of the function $g(x)$, making the necessary constant adjustments to obtain a logarithmic form:…
Find the indefinite integral of the function $g(x)$, making the necessary constant adjustments to obtain a logarithmic form:
$I = \displaystyle\int \dfrac{3x - 1}{x^2 - \frac{2}{3}x + 5} \, dx$
Integration by parts: reversing the product rule
When differentiating a product, $(uv)'=u'v+uv'$. Integrating and rearranging gives:
$$\int u\,dv=uv-\int v\,du.$$We work on an interval where the functions and derivatives used allow these operations. Choose $u$ so that differentiating it simplifies the expression, and $dv$ so that you can find $v$. The new integral should be simpler; if it is not, review your choice.
The degree of the polynomial decreases with each application: