Evaluating Indefinite Integrals

The process of evaluating indefinite integrals is essentially a quest to find the antiderivative, which is a function that, when differentiated, yields the initial function. This process is known as integration and is symbolized as ∫f(x) dx, where f(x) is the function we're integrating. The outcome, also known as the antiderivative, includes a constant of integration, C. This constant is included because the derivative of a constant is zero.

The problems about Evaluating Indefinite Integrals

Topic	Problem	Solution
None	Evaluate the following indefinite integral. \[ \i…	The integral is in the form of $\int \frac{a}{\sqrt{x}} d x$ , which is a standard integral form.
None	Evaluate the integral. \[ \int 9 y^{4} \sqrt{1-9 …	Let's start by identifying a part of the integrand that could be a good choice for a u-substitution…
None	Evaluate the indefinite integral. (Remember to us…	The integral is in the form of $\int \frac{1}{a x + b} d x$ , which is a standard integral form.
None	Find the particular antiderivative of the followi…	The given derivative is $C^{'} (x) = 4 x^{2} - 5 x$ . To find the antiderivative, we integrate \(C…
None	Find the following indefinite integral: \[ \int \…	First, we recognize that this is an integral of the form $\int u d v$ , where \(u = \sqrt{x} = x^{\…
None	Find $s (t)$ , where $s (t)$ represents the position…	Given the velocity function $v (t) = 3 t^{2}$ , we can find the position function $s (t)$ by integrat…
None	Find $f$ such that $f^{'} (x) = 4 x^{2} + 5 x - 5$ …	Given the derivative function $f^{'} (x) = 4 x^{2} + 5 x - 5$ and the initial condition $f (0) = 3$ .
None	Calculate the indefinite integral \[ \int \frac{4…	We are given the integral $\int \frac{4 d x}{\sqrt{36 - 9 x^{2}}}$ .
None	Find the indefinite integral. \[ \int \frac{x}{\s…	Given the integral $\int \frac{x}{\sqrt{x - 9}} d x$
None	Determine $\int\left[\left(\frac{1}{2}\right)^{x}…	The integral is a sum of two terms, so we can integrate each term separately.
None	Find $f$ such that $f^{'} (x) = 8 x^{2} + 5 x - 2$ …	We are given that the derivative of the function $f$ is $f^{'} (x) = 8 x^{2} + 5 x - 2$ and that the …
None	$\int \sin^{2} x d x$	Given the integral of $\sin^{2} x$
None	Find $f$ such that $f^{'} (x) = x^{2} + 2$ and $f…	The problem is asking for a function $f (x)$ such that its derivative is $x^{2} + 2$ and the funct…
None	$\int \cos^{2} (x + 1) d x$ أوجد قِيمة التكامل	$u = x + 1$
None	$\int tg x d x =$	$\int tg x d x = \int \frac{\sin x}{\cos x} d x$
None	$\int x^{4} d x$	Apply the power rule for integration: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$
None	[2] \( \int\left(4 \sqrt{x}+\frac{8}{\sqrt{x}}\ri…	$\int (4 \sqrt{x} + \frac{8}{\sqrt{x}}) d x = \int (4 x^{\frac{1}{2}} + 8 x^{- \frac{1}{2}}) d x$
None	Find $\int (\frac{3}{\sqrt{x}}) d x$	\int\left(\frac{3}{\sqrt{x}}\right) d x = 3\int\left(x^{-\frac{1}{2}}\right) d x