Find the indefinite integral by substitution.
{eq}a) \int x^3(x^4 + 3)^2 dx \\ b) \int 3x^2 \sin x^3 dx {/eq}
Question:
Find the indefinite integral by substitution.
{eq}a) \int x^3(x^4 + 3)^2 dx \\ b) \int 3x^2 \sin x^3 dx {/eq}
Indefinite Integral :
The indefinite integral of a function given in the form {eq}\int f(x)\ f'(x) \ dx {/eq} can be evaluated by using the substitution method. Since, the function and its derivative both are given as a single function
therefore, if we take {eq}f(x)=t {/eq} then we will get {eq}f'(x)\ dx = dt {/eq} and hence the integral becomes easy to solve.
Answer and Explanation: 1
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a) Let the given integral be {eq}I {/eq} then we have:
{eq}I = \int x^{3}(x^{4} + 3)^{2} \ dx {/eq}
Take {eq}x^{4} + 3= t {/eq} so that we get...
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Chapter 12 / Lesson 11Indefinite integrals, an integral of the integrand which does not have upper or lower limits, can be used to identify individual points at specific times. Learn more about the fundamental theorem, use of antiderivatives, and indefinite integrals through examples in this lesson.