Integration by Parts

The method known as Integration by Parts is a powerful tool in calculus, primarily utilized when dealing with the integration of functions that are products of one another. Derived from the product rule of differentiation, this technique simplifies the integral of a product into an integral that is significantly more manageable to solve. This approach is particularly beneficial when the integral is not straightforward to compute.

The problems about Integration by Parts

Topic	Problem	Solution
None	Evaluate. (Be sure to check by differentiating!) …	Given the integral $\int (4 t^{3} - 7) t^{2} d t$ , we need to evaluate it.
None	Evaluate. (Assume $x > 0$ .) Check by differentiatin…	Identify the parts of the integral that will be 'u' and 'dv'. In this case, let 'u' be $\ln x$ an…
None	Evaluate the indefinite integral. \[ \int 2 e^{2 …	First, we recognize that this integral is a good candidate for integration by parts, which is given…
None	Question 1. Graphical Integration Challenge: u-Su…	Let $F (x) = \int_{- 1}^{x} f (t) d t$ and $G (x) = \int_{0}^{x} g (t) d t$ , where the gra…
None	Question 1. Graphical Integration Challenge: $u$ -…	Given that $F (x) = \int_{- 1}^{x} f (t) d t$ and $G (x) = \int_{0}^{x} g (t) d t$ , we are…
None	Use the substitution $u = - x$ to evaluate $\int_{-2…	Let $u = - x$ . Then, $x = - u$ and $d x = - d u$ . Also, when $x = - 2$ , $u = 2$ , and when $x = 2$ , $u =…
None	Using integration by parts find $\int x \sin x \m…	Choose u = x and dv = $\sin x d x$ , then find du and v: du = $d x$ , v = -\(\…
None	$\int x^{3} e^{x^{2}} d x$	$u = x^{2}$
None	a) \( \int \frac{\operatorname{arctg} x}{1+x^{2}}…	Let $y = arctg x$