Evaluate the integrals in Exercises 87–96.
93. ∫₀^(π/2) 7^(cos t) sin t dt

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Recognize that the integral is of the form $\int_0^{\frac{\pi}{2}} 7^{\cos t} \sin t \, dt$, which suggests a substitution involving the exponent's inner function $\cos t$.

Let $u = \cos t$. Then, compute the differential $du = -\sin t \, dt$, which implies $\sin t \, dt = -du$.

Change the limits of integration according to the substitution: when $t = 0$, $u = \cos 0 = 1$; when $t = \frac{\pi}{2}$, $u = \cos \frac{\pi}{2} = 0$.

Rewrite the integral in terms of $u$: $\int_0^{\frac{\pi}{2}} 7^{\cos t} \sin t \, dt = \int_{u=1}^{u=0} 7^u (-du) = \int_0^1 7^u \, du$.

Now, integrate \$7^u$ with respect to $u$ using the formula $\int a^x \, dx = \frac{a^x}{\ln a} + C$, and then apply the new limits from 0 to 1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves setting a new variable equal to a function inside the integral, then rewriting the integral in terms of this variable and its differential. This technique is especially useful when the integral contains a composite function.

Exponential Functions with Variable Exponents

Exponential functions with variable exponents, such as 7^(cos t), require careful handling during integration. Recognizing that 7^(cos t) can be rewritten using the natural exponential function as e^(cos t * ln 7) helps in applying substitution or other integration techniques effectively.

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