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Use substitution to evaluate the indefinite integral ∫3x2e2x3dx. Solution. Here we choose to let u equal the expression in the exponent on e. Let u = 2x3 and du = 6x2dx. Again, du is off by a constant multiplier; the original function contains a factor of 3x2, not 6x2.
Write an integral to express the area under the graph of y = e t y = e t between t = 0 t = 0 and t = ln x, t = ln x, and evaluate the integral. In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.
In the following exercises, find each indefinite integral by using appropriate substitutions.
Chapter 9: Indefinite Integrals Learning Objectives: (1) Compute indefinite integrals. (2) Use the method of substitution to find indefinite integrals. (3) Use integration by parts to find integrals and solve applied problems. (4) Explore the antiderivatives of rational functions. 9.1 Antiderivatives Definition 9.1.1.
In this section, we explore integration involving exponential and logarithmic functions. The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, y= {e}^ {x}, y = ex, is its own derivative and its own integral.
16 Νοε 2022 · If \(F\left( x \right)\) is any anti-derivative of \(f\left( x \right)\) then the most general anti-derivative of \(f\left( x \right)\) is called an indefinite integral and denoted, \[\int{{f\left( x \right)\,dx}} = F\left( x \right) + c,\hspace{0.25in}\,\,\,\,c{\mbox{ is an arbitrary constant}}\]
Below is a table of Indefinite Integrals. With this table and integration techniques, you will be able to find majority of integrals.
